People in the UK don't all get the same standard of treatment - those with the means to navigate the system, pay to get into the first layer quicker or who have the freedom and the money to waste hours and days to access health care do better. (As you have done yourself) How is that different from people paying for extra insurance in the US?
You worry about calling ambulances? Urgent cases here are waiting not just a few hours but many. Patients in their 80s and 90s left lying in the rain and cold for several hours, dependent on the good will of passers by or neighbours to try and keep them warm whilst waiting (and this was happening before the pandemic, its just more widespread now).
You hope that PP upthread's dire appointment experiences were unusual? To me it sounded absolutely typical. Have you never tried playing appointment roulette with the many NHS fiefdoms? Every system is different and they don't communicate. If you are really lucky you get the details of the appointment before its happened and they have a record of it when you arrive.
Its equally likely the appointment arrives after the date, for the one week you asked them to avoid, or they have lost the record and don't have your details. You take a day off for an appointment at time X, the call happens at a random time when you are on the loo - tough luck. Down the snake you go.
I recently received a text for an appointment I hadn’t made (a very common problem with clinics which issue appointments rather than allow you to make them) - text came end of business day, lines closed. Spent two hours of my own time next morning trying to get through and then gave up.
There are hundreds of different appointment systems all setting their own rules and processes - there is no “one” NHS” and we should stop pretending there is.
Some of this is down to budgets but not all of it by a long chalk. A lot of it is bad business practices fragmented across thousands of little organisations all reinventing the wheel and often its bad procurement management. Our spend per head on health is not delivering the outcomes that Europeans get, with state backed and regulated health insurance models and hasn’t done for some years. And that isn't counting the huge number of patient hours wasted trying to navigate the process and waiting around for appointments. Patient time is assumed to be valueless.
It has also always been largely privatised, especially in primary care - that was forced by the doctors’ unions as the price of supporting the system.
The choice is not between a wonderful NHS and a dire US insurance system and people should stop pretending it is.
The model which was groundbreaking in the 50s is now out of date and has been improved on by most of our European neighbours who deliver more patient focused care at lower cost and seem to be better at enabling their professionals to get on with the job rather than leaving them buried in paperwork and using antiquated systems.