While I tend to believe that this is AT&T being corporate money-grabbing assholes, I have to disagree on the inference you made, that "we will allow the apps on our network if devs pay for bandwidth" implies "there is no capacity problem".
Charging for something is a way for regulating demand for a scarce supply of something. It's literally Econ 101, supply and demand. AT&T has to charge someone for the capacity used, such that the rates charged for it will regulate it. If there's a capacity problem, the rates go up. When the rates go up, demand goes down, and the capacity eventually reaches equilibrium based on price. It's how any producer sets the price of something in limited supply and high demand.
If demand is high enough for a sustained amount of time, then it's in AT&T's best interest to expand the production capacity (i.e. increase bandwidth available on their network), thus raising the supply. The marginal price goes down, but they are selling more total bandwidth, so their total revenue goes up. If they don't expand in a timely manner, a competitor comes in with better service for the same price, and all AT&T's customers leave and join the competitor.
In any case, you need to attach a price to the thing in limited supply so that it self-regulates. If no one pays for it, that's when there's a capacity problem.
If you want to argue about AT&T selling unlimited data plans that aren't really unlimited, that's one thing. You can also argue that bandwidth is not a true "physical" resource that takes cost to produce; once a certain capacity is in place, you shouldn't charge for usage. You can also argue that spectrum itself is scarce and the government grants a monopoly to these few companies, so competition is limited or nonexistent, and so they should be regulated. These are all fair arguments. But the general inference of "devs pay for bandwidth" => "no capacity problem" is fallacious.