We've presently got bigger revenues than City, and can possibly sustain that while continuing to compete in similar competitions, so it's highly unclear we can't compete with City on signings.
The PSG problem is trickier: they can work with a relatively lop-sided and small squad stacked with high wage stars, and still rest key players flexibly due to their weak domestic opposition. I can see us having trouble with them again in Europe with comparable budgets, due to their fewer injuries and better legs at the end of each season.
As for the present, I reckon City are stocking up on players because they've got an idea of some sanction that's incoming.