Tell me you've never needed significant healthcare without telling me.

It's never about "ambulances". The bureaucracy will happily call an ambulance for you, as it ends their involvement and absolves them of liability. And the ambulance company is happy to show up, as it's a large billable event. In fact when seeking/obtaining complex medical care, this is one of the failure modes you have to constantly swat away!

As for interacting with the bureaucracy to obtain real care, it's gotten to the point where it's easier to get something done calling a large bank. Navigating the medical bureaucracy is probably on the level of calling Comcast-Warner (although I haven't had the misfortune of having to deal with them for a decade, so I'm sure they've gotten worse as well).

Even before "AI", my standard experience is a multi-level phone tree, multiple mandatory announcements before you can even proceed, then waiting on hold. That merely gets you to an answering service who types a ticket into the computer system. Then you wait a ~day for a nurse to call you back (if you miss answering this call, return to step 1). Then the only urgent appointments are invariably with "nurse practitioners" who throw pre-canned advice at you, as they don't really know what they're doing. That's four layers running interference to keep you away from an actual doctor, who you will finally get to see in a month for a fifteen minute appointment, if you are lucky.