Solar Is Cheapest Electricity in History, U.S. DOE Aims to Cut Costs 60% by 2030

This is more or less what post-combustion capture carbon-capture-and-storage plants do, with the minor detail that they do the actual combustion above ground instead of below ground. But the inputs and outputs are as you describe: oxygen from the air, fossil fuels from underground, CO₂ outputs injected back underground, heat outputs sent to somewhere unspecified, typically the air.

CCS power stations are substantially more expensive than the usual kind of fossil-fuel power station, and they are generally considered to be economically uncompetitive.

It's probably better to do the actual combustion aboveground in many cases—although it makes your power plants easier to blow up with bombs, it also makes them enormously easier to build and maintain, and much less dangerous to work in when nobody is trying to blow them up.

Economically uncompetitive compared to solar/wind 500% overbuilt? How many hours storage have to be added to the solar / wind to make a ccs gas plant competitive?

Probably, yeah. As I said in my other comment, the whole question of utility-scale storage cost modeling is difficult to get a good handle on.