We can't be held responsible for their own financial mismanagement - likewise they should not be eligible for any bailout funds - or they can sell their stock on the market to raise funds.
Many many years ago Air New Zealand essentially went bankrupt (after it had been privatized) so the NZ gov bought it out. Then a few decades later privatized it again at a profit.
Which is the evidence that this can be done.
Why not though? If they go under, someone will step in to buy the assets and hire the employees back. It's not like air travel will go away forever.
They don't need to be indispensable. they just need to be critical enough that not bailing them out is much more expensive than bailing them out.
The solution is to prevent this scenario from ever happening in the first place, by either requiring sufficient competition, or requiring sufficient capital reserves (see: banking and reinsurance), or just nationalizing the thing.