Does the Bitcoin Network still have a way of notifying nodes of a potential fatal bug/error?
No.
After all, it is a peer-to-peer network without any central control. Either no-one can issue alerts or everyone can, including the malicious. Changing this would go against the founding principle of Bitcoin - avoiding any need for trusted third-parties. It seems to me the removal of the alerting mechanism moved the Bitcoin network closer to its founding principles.
The discussion of this feature is illuminating
The alert system was a frequent source of misunderstanding about the
security model and 'effective governance', for example a years ago a
BitcoinJ developer wanted it to be used to control fee levels on the
network and few months back one of Bloq's staff was pushing for a
scheme where "the developers" would use it to remotely change the
difficulty-- apparently with no idea how abhorrent others would find
it.
The system also had a problem of not being scalable to different
software vendors-- it didn't really make sense that core would have
that facility but armory had to do something different (nor would it
really make sense to constantly have to maintain some list of keys in
the node software).
It also had the problem of being unaccountable. No one can tell which
of the key holders created a message. This creates a risk of misuse
with a false origin to attack someone's reputation.
Finally, there is good reason to believe that the key has been
compromised-- It was provided to MTGox by a developer and MTGox's
systems' were compromised and later their CEO's equipment taken by the
Japanese police.
If not, what is done if the Bitcoin Network is compromised?
Presumably lots of different people do different things while attempts at coordination are managed through media popular with Bitcoin developers.
You could look at what happened in earlier crises such as the Value Overflow incident of 2010. Nowadays it would probably be much harder as there are a greater variety of software applications in use. I don't know if it can be argued that the Value overflow incident was a bug in one software application rather than a bug in the network protocol. In which case it isn't a good example. Another example might be the removal of vulnerable or unused opcodes from Bitcoin script because of the OP_LSHIFT bug
See https://www.cvedetails.com/vulnerability-list/vendor_id-12094/Bitcoin.html. This is perhaps a clue to how discovered vulnerabilities are handled. There may be little advantage in having a network specific mechanism. Certainly the history suggests that securely embedding an alerting mechanism in a purely peer to peer network protocol is a difficult problem to solve.
