Well what if the fees are not enough, or there are not enough volume tx going on, or anything else?
Then mining difficulty will drop, until remaining miner can mine at profit with lower difficulty.
Would non-monetary use cases like Ordinal inscriptions solve the problem?
No, unless block usually isn't full.
Obviously, the top priority should always be monetary transactions, then if that wasn't enough, the incentive could be filled with non-monetary transactions. Perhaps at the protocol level, make it so that monetary transactions go first, then with less priority process non-monetary transactions. We should consider if ordinals can be saved or not long term, if these use cases could deliver something useful. If they will only bring problems, then perhaps a full frontal attack on them and planning to stop them may be best. But im still open to discuss this before I pick a side.
With current way of Bitcoin protocol, miner can choose any transaction they want to include in a block. In practice, miner usually include as much as transaction as they can starting from transaction with highest fee rate. Forcing miner to prioritize certain transaction would be difficult for various reason, such as each node have slightly different mempool.