There is a tradeoff here which might mislead. That is that when you're wirelessly recharging a battery "good" is defined by "charges in a short time" which means a lot of current and thus a lot of heat. When operating a device "good" is defined by "runs a long time" which means minimal current and thus little heat.

So before we could dismiss the idea, we would really need to run the experiment of "running" via wireless power to see how the current requirements in that state would be reflected in device heating (or not). I agree with you though that if the answer was "always smoking hot" then it is a non starter.

As for size, the "wireless" part is really really thin (a loop antenna). Battery size is a function of how much electrolyte it can carry and LiOn batteries are about 0.5g/cc and 250Wh/kg (or .25Wh/g) iPhones are around 12Wh so about 48g of battery which is 96cc of Lithium. Call it 100cc and you're looking as a "7.8 x 15 x 1" cm "battery card" (order of magnitude approximation based on that math and rough measurements of my iPhone 13SE which is a smallish phone)

I like the idea of magnets but I don't like the idea of the battery being "jostled off" my phone in my pocket and so effectively turned off when I think it should be on. So I think that would take some design work to get right.