A very recent example from me, I had my first appointment at an NHS physio scheduled 4 months after my shoulder dislocation, which is a full month after I already finished recovering by going to a private physiotherapy clinic.
after I already finished recovering by going to a private physiotherapy clinic.
So, you're saying the method generally used in the US worked for you, and treated you faster.
Unfortunately, at a cost. But I'm pretty sure that's the way it works.
A same-day appointment to a family clinic (Of which you'd already have an association with - I only say it like that because I don't know of a similar practice in other countries) would get you in and out within an hour. Cost is dependent on your insurance, but probably a $30 or less (even $0) visit.