> people will start directly using these currency for payments (which they already do) without exchanging to fiat

Without fiat exchange, People can only accept crypto in an economically feasible manner as payments up to the rate that they have expenses/outflows that can be paid in crypto. The price will only be able to stabilize at a price that is orders of magnitude lower than today. How much lower exactly depends on the health of the black market for exchanging with fiat.