I used to be an ancap, and I want to help lay the groundwork for helping you to understand why I've moved on to anarchism, putting things in terms you're more comfortable with than what you're probably used to seeing.
The first thing we need to discuss is property. It's important to recognize up front that property is a social convention. It's an extremely important one, possibly even the most important one, but it is still essentially a social convention. Property has been formulated in different ways by different people at different times. This establishes a critically important fact about it, which is that what constitues valid property can be changed. The rules on how you can become the first owner of property, what can be property, and what an owner can do with their property as property can be drastically different from what it is today.
It's very likely that what we have today isn't your ideal form of property. You may think that intellectual property isn't valid property. You certainly believe that state property isn't valid property. You probably think that claiming unimproved land as property without even fencing it in isn't valid property. If you do belive in intellectual property, you may still believe that the owner has no right to exclude others from their ideas using force, and only has a protection of their identity as the originator of the idea against fraud. Whatever specifically it is that you think about property, recognize that these are essentially social conventions, not absolute facts, and that the reasons that you give for thinking that intellectual property or unimproved land property are or are not property are based on problems that you identify with the alternative, and not on their being absolutely true or false. Someone who opposes intellectual property is not saying that lack of enforceable property claims would have no effect on the development of ideas, nor is someone who supports intellectual property saying that there is no value in letting ideas spread and be used freely. Whichever you think is the bigger problem likely determines which side you'll fall on, and this is shaped more by your perceptions and framings of the situations being discussed, shaped by your history and experience. It is also socially negotiated with others (at times, indirectly through the state, at other times, more directly).
The next thing we need to discuss is the market, and what the market means. To do that, we also have to discuss what the market is not. A common framing of the market among ancap thought leaders is that the market is "the sum of all voluntary human interaction". This isn't what most people mean by it, though, they mean exchanges of property (or promises, through contract, but for brevity I'll consider them together as "property", they are similar enough social constructs for how I'll use it here). Helping someone at the side of the road change their tire is not a market act, unless it is done with an exchange of property. Giving in charity is not the market, because it is irreciprocal between giver and receiver, regardless of whether you are exchanging money in hand for a feeling of moral superiority or satisfaction in being a good person. Going to the bar with friends is not a market interaction between you and your friends. Sharing property in marriage is not a market interaction between you and your spouse. If we're going to use a definition of "market" that is broad enough to cover all these situations, you should expect that you will be misunderstood, and you should expect that anyone suggesting the market isn't the solution to all problems doesn't mean the same thing by "market" that you do. Even those who call themselves "voluntaryists" often fall victim to this thought pattern.
Considering that most people understand markets as essentially being about property, it's important to recognize that this means that markets cannot be any less a social construction than property itself is, and property itself is essentially a social construction. This doesn't mean that markets aren't important, any more than it means that property is not important. What it means is that both property and markets are much more flexible as social constructs than they are in the way that you ordinarily think about them. We no longer recognize the slave trade as a legitimate market because we no longer recognize ownership of people as property to be a valid form of property. We no longer recognize debts (a form of property, in the expansive sense) as inheriting from parent to child. These changes in property had transformative effects upon the markets that were retroactively made into non-markets.
The third thing we need to discuss is the word "capitalism". In your mind, "capitalism" refers to free markets with private property. Again, this isn't how other anarchists use it, but I'm talking to you in how you use it here. Noting what has already been said about property and markets both being socially constructed and thus socially changeable, this would leave "capitalism" in a spot very open to forms that don't look at all like what we typically imagine as capitalism. This should be fine by you though, as the capitalism you seek likely looks very little like that capitalism that exists today. The capitalism you want brings so much possibility and potential, that it would be small-minded to think it would be like today, except less government. It can be so much more than that, so much better than that.
But with private property, like all forms of property, being socially constructed, you may have beliefs of your own on what constitues valid property and why, but this is still something that needs to enter into the social dialogue and become accepted for your model to prevail as the actually enforceable form that property can take. If the social system accepts property as you formulate it, rejoice in your success, but recognize that if the social system rejects property as you formulate it, it is no infringement of your inalienable rights, rather it is a suboptimal formulation by your standards and reasoning of privileges that are granted to you by the social system. Property rights aren't absolute in the same sense that personal autonomy can be, and if property can be formulated in such a way that it is never a violation of your autonomy, and only a different understanding of the incentives and structures in which you can negotiate for what is yours, then the discussion on property can move on to a new level of maturity that you likely haven't seen it at yet.
"Voluntaryists" take note, that this applies to you as well, even if you do not cling to the word "capitalism" as tightly for all the confusion that the word creates and all of the problems that people recognize in the current system. If you define voluntariness in terms of property, you're defining voluntariness in terms of something that is a malleable social construct. The terms of voluntariness can be changed, in that sense, and it's weak conception of voluntariness that is so subject to change. For as long as voluntariness is tied specifically to property, the strength of your argument is limited. A conception of voluntariness that is fundamentally rooted not in property but in autonomy is the strongest conception of voluntariness.
Now, is there such a conception of property that, whatever the rules of property are, it will never violate your autonomy? Coming up with examples that do violate your autonomy is easy to do, even trivial. If you can be enslaved to pay your debts, that is a clear violation of your autonomy. If you can be enslaved to pay your parents' debts, that is a clear violation of your autonomy.
If you can be killed, kidnapped, or enslaved in the name of property, promises, and debts, then you are in a property system that can and will violate someone's autonomy, probably even yours at some point. What would a conception of property look like which would never violate your autonomy?
Such a system would have the potential to bring "left-anarchists" and "right-anarchists" together under one larger tent. To limit one's conception of property in such a way makes the communist powerless to subject the capitalist to their whims, and the capitalist powerless to subject to communist to theirs. It is only because both often think that the other's conception of property requires aggression against them, and in such a way that violates their autonomy, that this is even a problem. What if we can get past this, by all factions rejecting any conception of property if it is found that in that conception of property, one's autonomy would be violated?
If you believe that intellectual property is legitimate as property, but find yourself operating in a social system that doesn't recognize intellectual property, then if you know that your intellectual labors will not be rewarded in the same way as if they were protected by intellectual property, it is no violation of your autonomy that people reject the mechanism of reward you would like to have, you can simply choose whether to dedicate your efforts elsewhere than in producing such intellectual products, or produce them only to the limited extent that you find the rewards for your efforts to be fulfilling.
Consider this effort to develop a system of property, possibly even a range of property systems, in which property and its adversarial enforcement can never violate autonomy, to be the project of developing an even more deeply libertarian property theory than libertarianism has today.
Consider that property is not a fixed idea, but just another part of the freely negotiated social system of the post-state future, that it is likely to change over time, and that such changes may bring improvements as substantial as the changes of the past (e.g. abolition of slavery, debt peonage, and inter-generational debts).
Consider that when you hear left-anarchists oppose property, they are not opposing property in general but facets of its currently-existing institutional form. When Proudhon said "Property is theft!" he did not mean all ownership was illegitmate, but that property was the justification given for massive injustices he saw around him.
Consider that when you hear left-anarchists praise democracy, they are not praising the rule of one group over another, but the decisionmaking process by which the members (read as: co-owners) of the social institutions that they form together decide how best to achieve their common goals, that they often find voting to be a last resort when other methods of decisionmaking fail, and that they would often sooner see a democratic unit organically separate over irreconcilable differences than see one party overpower the other.
Then reconsider whether the fullest expression of anarchism is capitalism in particular, whether expressing that to be the case is valuable, and whether there is enough common ground to be found when those elements of capitalist rhetoric most likely to create friction and confusion are left behind. If you reach the same conclusion I did after considering these things, you may find that "ancap" is no longer a useful distinction for you.