Why would a third-party be in your product's critical path? It's like the old business school thing about "don't build your business on the back of another"
The reason third-party things are in the critical path is because most of the time, they are still more reliable than self-hosting everything; because they're cheaper than anything you can engineer in-house; because no app is an island.
It's been decades since I worked on something that was completely isolated from external integrations. We do the best we can with redundancy, fault tolerance, auto-recovery, and balance that with cost and engineering time.
If you think this is bad, take a look at the uptime of complicated systems that are 100% self-hosted. Without a Fortune 500 level IT staff, you can't beat AWS's uptime.
E.g., a hospital could keep recent patient data on-site and sync it up with the central cloud service as and when that service becomes available. Not all systems need to be linked in real time. Sometimes it makes sense to create buffers.
But the downside is that syncing things asynchronously creates complexity that itself can be the cause of outages or worse data corruption.
I guess it's a decision that can only be made on a case by case basis.
i bet only 1-2% of AI startups are running their own models and the rest are just bouncing off OpenAI, Azure, or some other API.
Good luck naming a large company, bank, even utility that doesn't have some kind of dependency like this somewhere, even if they have mostly on-prem services.
"Some kind of dependency" is fine and unavoidable, but well-architected systems don't have hard downtime just because someone somewhere you have no control over fucked up.