Classical mining is based on finding the smallest possible block hash. But as we know, r value of the signature can be smaller than usual if it starts with some zeros. Leaving half of the base point behind, I can imagine a system where miners collect transactions and try to find some value that can be added to "r" and can lead to smaller signatures. Users can receive such values in P2P way, because each miner can just share his "r2" public key to prove that miner found the lowest public key for that signature. Then, that user can send some satoshis in LN as a reward to that miner, and to unlock that coins, the miner have to reveal that private key for "r2" to the user. Then, that user can create smaller signature, leaving more room for other transactions in the blockchain.

Assuming one satoshi per byte fee rate and 2,000 transactions in the whole block, saving 5 bytes per transaction would mean earning 10k satoshi per block (and leaving some room for 40 more transactions 250 bytes each). Of course it is negligible for big miners, but for smaller ones it may be profitable, if they could fight for some satoshis. And the profitability limit will be quite high, because truncating five or six bytes may be profitable, but hitting eight or more will be exponentially more difficult, so it rather encourage hitting more transactions than more zeroes in single transaction.

It can be applied to any address type that involves public keys, including multisig and Segwit, because transaction fees are based on total transaction size. So, even if Segwit signatures are outside of the block, users can still pay less satoshis for including such transactions with "mined r values". Unfortunately, that kind of service cannot be applied for taproot, because that signatures has fixed size of 64 bytes, but they are compressed to single signatures anyway, so there is less need for that kind of compression.