"Doctor's surgery" is what we'd call the doctor's office in the US, right?

In terms of walk-in, there's "urgent care" and similar things. And of course the emergency room if the need is severe. Telehealth was vastly expanded during the pandemic and has stuck around to a significant degree.

In addition to urgent care clinics, every doctor I've had holds a few appointment slots open for same-day or next-day visits. I just have to call them when they open.

Just checking my doctor's group, they have multiple appointments available for new or existing patients in all 7 of their locations around Austin as early as Tuesday - and that's probably only because Monday is a holiday. The only caveat is that you have to accept whichever doctor is available, rather than selecting you PCP.

I know it's a big country with a lot of variability of experience, but this has been the case everywhere I've lived.

Yea, we basically use urgent care for everything, since it seems very few in-network doctors want any more patients, and the ones who do have no availability for months. So we're in this weird position where we have no "primary care doctor".