short answer: yes, you can do anything you described with the Bitcoin integration.
ckBTC is just one possible solution where the canister still controls the Bitcoin UTXOs and mints/burns ckBTC on deposit/withdraw.
anybody could build sth. similar to that. for many dapps on ICP it makes sense to just utilize ckBTC for simplicity and convenience. why reinvent the wheel if not needed?
a canister can also control UTXOs without exposing any information at all and without minting a “wrapped” version. recently I started a discussion about that because I think it would be great to have a standard that allows canisters to proof that they control a specific address on other networks, see Standard to expose verifiable TVL of assets secured by ICP (Chain Fusion)
some references: