Shared Coinbase transaction typically is 32 kB data (an average of 4500 items) and doesn't need any further verification, like checking UTXO, mempool, whatever.
Since PoW should be considered an essential part of the header, what you are proposing then is to increase header size from 80 bytes upto 72 KB (worst case 10000 items), a nearly 1000 fold increase...