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...
This is more significant when considered in conjunction with the 0.02 * threshold on finishing a block. That threshold means its more likely that two blocks will be finished closer together than for 10 minute block periods and thus the increased propagation and verification (for the up to 10,000 block solutions) can be significant relative to the spacing between duplicate finished blocks. As I wrote in my prior post, all of this contributes to amplifying the selfish mining attack.
Well, @tromp is not on the point, neither you @anunymint:
The Shared Transaction Coinbase is not a part of the Header, its hash(id) is,
The transaction itself is part of the block, like conventional coinbase transaction and other transactions. The block size remains the same as what protocol dictates, plus the size of this transaction it implies an almost 5% increase (worst case) which is not a big deal.