Using a capacitor instead of a coil for wireless charging?
First point: A transformer has a built-in return path, so you have a complete circuit. That is to say, it has two wires on each side, so current can flow out through one and back through the other. To make an equivalent device with capacitors, it would be need to have two of them, one for out and one for return.

Next lets put some rough numbers on.  For a phone, you have an area of about 5000mm^2 for each capacitor. If you put the capacitor plates near the surface, then you might get a gap of 1mm and the plastic case could be engineered with a dielectric constant of say 10. That works out about 0.4nF per capacitor.

Now lets think about transferring power through that. At 1kHz the impedance would be 400Mohm. At 1MHz, it would be 400kohm, and 1GHz it would be 400ohm.  That sort of high frequency causes lots of headaches around avoiding interference with other devices. And at any real power, not microwaving the user.  And even if you go all the way up to 1GHz, then you still have to deal with a relatively large impedance. To get a watt flowing through the capacitors will need 30V.

So the engineering problems are probably not insurmountable, but it is still much easier (and therefore cheaper) to use an inductive method.