I think you're misunderstanding what the Request network is doing. Their goal is exactly what you described, a payment gateway for existing (all) cryptos (including fiat). The Request token ($REQ) is only used to collect fees for using the request. The REQ tokens are burned as part of the transaction, and in the end goal the end user themselves won't have to actually ever buy any REQ tokens either, it will automatically be done by the network. REQ tokens are actually only ERC20 tokens (meaning they're using the Ethereum blockchain).
The end goal is a decentralized payment gateway that will accept any crypto or fiat and payout in the desired currency. Request will handle the swapping of currencies and everything else in a decentralized manner. Obviously, likely quite a ways away from that, and there are a lot of scaling issues that need to be solved with blockchain tech itself before this can actually exist in a usable way.
The end goal is a decentralized payment gateway that will accept any crypto or fiat and payout in the desired currency. Request will handle the swapping of currencies and everything else in a decentralized manner. Obviously, likely quite a ways away from that, and there are a lot of scaling issues that need to be solved with blockchain tech itself before this can actually exist in a usable way.