If a serious issue like this was ever found in the Bitcoin Core client again, would the offending txs be removed again? if so, then why doesn't this happen when bugs are found in other clients?
Does Bitcoin Core have some kind of privilege in this sense? or was this just a once off thing, or did the small bitcoin community at the time manage to get everyone to agree to have this tx removed?
I think you're looking at it wrong.
The blockchain forked when the transaction creating the 184 billion BTC was mined. "Correct" clients will have rejected that block and started building their own correct fork, while Bitcoin Core went off on an invalid chain.
Fixing the Core code and forcing it back onto the correct chain resolved the fork that was caused by the bug; it didn't create a fork.
It's possible that nobody found a block using anything but code derived from Core, and so the correct branch didn't get to diverge from the incorrect branch, but that's beside the point.