I don't know if this would achieve anything or be useful but is it possible to make pow and pos tokens on layer 2 of the bitcoin blockchain (in a way they'd have their own mining and compressed/hashed versions of the blocks and certain transactions could be entered into bitcoin's blockchain but most could be stored elsewhere).
I think that something like this already exists on layer two sidechains like Liquid that is made by Blockstream, and maybe some other chains like RSK.
They have tokens built on top and option for more privacy with Liquid Confidential Transactions, but I think that additional mining would not be very popular of profitable.
One problem I see with this is less decentralization because Liquid is federated network, so you would have to trust them and have less decentralization.
Do you mean something like the
Drivechain project, started by Paul Sztorc?
I heard about Drivechain but it never received much attention from Bitcoin community.
Another possibility which would be even more independent from the Bitcoin chain would be an algorithmic bitcoin-based stablecoin on a chain with smart contract functionality (doesn't have to be Ethereum-based). Such a chain could however not directly interact with the Bitcoin chain (only via atomic swaps).
Stable coins are also starting to show up in Lightning Network and they are totally backed by Bitcoin, so there is no connection with ethereum.
I am not a fan of any stable coins, but they can be useful for temporary use case, and Lightning Network could be used for improving privacy, if done correctly.