So, does the current version of the Core mining software accept transactions with those NOPs, for inclusion into the candidate block?
No - it would not (as it could not know whether or not they are valid).
Would that software accept as parent a block mined by some other miner that contains them?
Yes it would as it assumes that whoever has mined them did know that they were valid.
Are you starting to get it now?
(fingers crossed that the penny has dropped)
Also without relaying you actually wouldn't have a P2P network at all (so it is an essential part of the system not some optional thing) and there are expectations that a node will not relay invalid txs or blocks (and any node that does will be banned by your client).