Since the content of the block is already selected by the time the block header is constructed, yes, theoretically a cryptocurrency could commit to the UTXO set in its block header. It would be more realistic to put such a commitment in an output on the coinbase transaction where it would not require a hardfork, though. As long as there are clear rules how to create and validate such a commitment, this commitment could also just cover a subset of the UTXOs, although I don’t see an advantage to committing to less than the whole UTXO set.

Since the content of the block is already selected by the time the block header is constructed, yes, theoretically a cryptocurrency could commit to the UTXO set in its block header. For Bitcoin, it would be more realistic to put such a commitment in an output on the coinbase transaction where it would not require a hardfork. As long as there are clear rules how to create and validate such a commitment, this commitment could also just cover a subset of the UTXOs, although I don’t see an advantage to committing to less than the whole UTXO set.