Here's a fuller illustration of that toy model of a market to more completely illustrate how the existence of rent breaks it.
This market consists of two identical people, one kind of consumable good (food), one kind of capital (land), and one kind of labor (working the land to produce food). There is exactly enough land to produce enough food for normal consumption for two people. The two people agree on what the rules of acceptable behavior (laws) and division of property are, so we don't need to worry about external law enforcement here. We will explore a couple of different scenarios of different laws and different divisions of property.
If the land ownership was evenly divided, the normal state of affairs would be for each person to work his land to produce food for his own consumption. Trade between the two people could be possible still -- one could give the other some of his food in exchange for the other taking over some of his labor, or they could divide the labor up into different types (planting, watering, harvesting, etc) and each do that for the other in exchange for some quantity of food -- and if the law includes laws against violence, then neither can coerce the other and all their trade would be free. Sounds wonderful.
But now, say one of the people (Alice) only owns 1/4 of the total land, while the other (Bob) owns 3/4 of it. Alice now cannot produce enough food for her own normal consumption, whereas Bob could produce an excess. Of course that means Alice is also working half as much as she would, and Bob is working one and a half times as much, so maybe that seems fair, but there are now pressures at work that will quickly make it unfair. Alice isn't eating enough, and Bob has more land than he really need to provide for his consumption, so Bob offers Alice a deal: "I'll let you borrow my extra land to work, in exchange for just half of its normal yield." So Alice gets some more food, and Bob gets more food too so he has to work less, and they both win right? So this is an obvious no brainer for both of them, a totally voluntary, free market transaction, right? Everything's great?
Except that now Alice is working a normal full work load but only getting 75% of a full crop from it, and Bob is getting a full crop but only working 75% as much for it. And this will continue indefinitely, and never end. Just because Bob started out with more than Alice, Alice is now stuck permanently getting less for her work than Bob, and Bob can permanently live a life of leisure on the back of Alice. Not because one of them did anything to win this position, not because one of them is better than the other, because remember we stipulated that they are identical; it's just because one of them began with more capital than the other.
But wait, it gets worse! Alice is still not eating enough, and desperate for anything that she can to get more. And now that Bob only has to work 75% of his land to meet his own consumption, he's got another quarter-field now lying around that he could lend out to Alice as well... in exchange for part of its normal yield again, of course. Which means Alice is now working even more than a normal full load and still not getting a full crop, and Bob can work even less and still get a full crop. Which means Bob now has another portion of his field he's not working anymore, and Alice still isn't eating enough, so she'll gladly work that too, in exchange for a part of it's yield sure, she needs the food so it's better than not. The endgame of this progression is that Alice has to work both fields in full if she wants to get a full crop to herself, meaning Bob doesn't have to work at all, still gets a full crop to himself, and Alice is his slave.
(Worse still, if Bob was really smart and Alice was desperate enough to fall for it, Bob could keep working full time, trade the excess he gets from Alice back to her in exchange for her land -- bought, not borrowed -- and gradually own even more and more of the land and make the whole scenario above worse and worse. Bob's ability to leverage his capital advantage to produce an excess of product allows him to increase that capital advantage further and further and accelerate the whole process -- at the accelerating expense of Alice).
Now, consider if the law was that you couldn't create those kinds of borrow-at-interest obligations, i.e. you couldn't rent. That's not to say that Bob couldn't let Alice use his land, or that Alice couldn't give Bob some of her crops; just that Bob couldn't oblige Alice to give her crops in exchange for use of his land. Well, you might say, that's just going to mean Bob won't let her borrow the land, and Alice will be permanently starving, and half a field will lie fallow since Bob doesn't need it. But wait! Isn't there an obvious, free-market solution to both of those problems, that benefits both of the parties? Bob could sell his extra land to Alice! Alice gets the land, and in exchange owes Bob some large amount of food (so Bob doesn't have to work as much to grow his own). Of course she'll have to pay that over time because she doesn't have enough to feed herself as-is. And of course even with the land, Alice will only be able to produce enough food to feed herself, so she will need to get more food to give Bob from somewhere else... meaning, from Bob, which she will have to trade her labor for, giving Bob even more leisure at no loss of consumption.
So for a while, you get something that looks just like the terrible outcome of the second scenario above. Alice works more than a full load just to keep a full crop at the end of the year, and Bob gets to slack off and still get a full crop to himself. But unlike the scenario above, this is temporary; with every year that Alice is working extra to pay off the land she bought from Bob, Alice gets more and more equity in the land, and Bob gets less and less. And eventually, Alice owns a full field to herself, and Bob only has a normal field with no excess. And Alice can stop working extra to pay off Bob, and Bob has to start working a full load if he wants to keep his full crop. And then we're back at the free, fair, and equitable arrangement from the first scenario. We have gone from a capital inequality to capital equality, redistributed wealth from the rich to the poor, all through completely voluntary transactions, with no coercion!
And all we had to do for that to happen was to agree that one person can't be obliged to give someone something in exchange for merely being allowed to use something else. Exchanges have to be proper trades; if someone is going to owe you something, like Alice owed Bob food, then you have to owe someone back, like Bob owed Alice land. Just do that, and the natural mechanisms of the free market will automatically move capital from the rich to the poor and bring everyone back to equality. But allow rent like that, and the free market breaks completely, wealth concentrates, and pretty soon the poor are slaves to the rich, and everyone turns the other way because every step of it was "voluntary", including the rental contracts.
But we don't allow people to "voluntarily" sell themselves directly into slavery -- such contracts are invalid -- so why would we allow contracts that indirectly tend toward the same result, when we could just consider them invalid and then let the market sort itself out?