First of all, I don't think they are many people who do not rely on bitcoin core professionally. Some run a different implementation for personal use but when money is at stake the risk of loss is too high. Even if it is a small bug that makes you miss a transaction, it could be a very large transaction that ends up costing you your business.
So alternate implementations are mostly for side projects, and even if they fork, their impact on the consensus is too small to provoke a split. In fact, the current opinion is that none of the alternate implementations are fully compliant with bitcoin core (https://bitcointalk.org/index.php?topic=923409.0). In deciding what to do regarding a potential fork, I think we should ignore the impact on alternate implementations.

Since you talk about a bug in bitcoin core, it is a different story altogether. Let's say someone found a major difference which would split the chain. If only client nodes are affected, it would mess up network propagation. If miners are affected, it may split the blockchain. Most likely, people will lose money and they don't care whether it helps fixing bugs or not. I wouldn't care either.

I don't think the damage on the credibility of bitcoin is worth the extra kick in the anthill though I have the opposite opinion if we were talking about an altcoin.