Questions upgrading wireless in the house to get wifi through my whole house

A little background: I have a 1600square foot ranch house. The problem is that my internet comes in from a cable that is run to a storeroom in my basement, all the other cable lines in the house are dead. Right now I have an old wifi n router in the basement with cables run to the office (also in the basement), and to 2 room on the main floor. The two rooms on the main floor are on the same side of the house. One of those cables runs to a wifi access point. The unwired side of the house is all bedrooms and

Right now my wifi signal sucks. especially in the end of my house opposite where the wifi access point is.

I can't easily run new wires.

I have a budget of $300.

Is it worth spending that much money on an wireless ac system when ax is now out? ax is still very expensive and I do not have fiber, my speeds are usually around 200mbs down. with those speeds i assume its not worth spending the extra for the newer wifi protocol.

would it be better to have a wired router in the basement and just have wifi from a single powerful ap on the main floor? or even have a wifi router on the main floor and connect it to a switch in the basement that connects to the other wired connections?

or is there a single wifi router that I can have in my basement that would manage to give me coverage?

I looked at a few mesh systems but they never seem to have enough wired ports on the main router.