I think it would be hard to make a decentralized swap exchange that runs between various chains.
We need some kind of bridge to connect between the chains, for example between Ethereum and Binance smart chain network.
And if there is one chain that have high transaction fee, then our cost to swap between chains will be high too.
Basically, there are many things to consider when we deal with different chains and networks.
This can be work, if the engineer is good enough most probably he can deliver the required network of the team and the project. You can operate DEX without having your own token but I believe this can only work at first because later on, you’ll realize the importance of having your own token and that could be your next big project, to introduce your own token.
i think it could work as long as this dex run on certain chain as use its token as transaction gas fee. dex token has no correlation with dex operation, it used only on its governance not as fee. uniswap be an example , this dex token that called UNI not used on each swap but using ethereum. so its the concept dex running on certain chain and dex utility for community itself.