The point is that in your analogy you assume two things that I summarize as competency, you assume that the medical staff is making the proper diagnostic and that from that diagnostic it is following the proper procedure. Based on these assumptions you conclude that getting quickly worse after the procedure is normal and predictable that is fundamentally wrong, it's only normal and predictable if the diagnostic is correct, if the procedure is correct and if the procedure has been correctly performed.
Now that's not a great analogy when it comes to managing a Football team because medical protocols are a lot more tight and universally understood to a high standard. Football managers have a wide range of competency levels even among the ones that have had a bit of success that's why while it makes sense to give a high degree of benefit of the doubt to medical doctors, it's borderline irresponsible to do so with managers, the vast majority of currently successful managers won't be successful if they change club, the vast majority of them aren't competent enough to coach at a higher level than their current level and never will. And even the ones that have been good or are still good at the highest level sometimes mess up badly.
All that to say that managers should be judged on what they are doing right now, not what we imagine time will do in their stead, not on what we imagine may happen in an alternate reality where everything is different and we should never assume competency on any manager that isn't already at the very top of Football and has been for a while. Amorim may end up being good or even great but until he actually shows something with United, we shouldn't advocate vehemently for time, we shouldn't assume that he is competent(or incompetent) and we need to stop this BS with necessary pain.
For reasons that can't be explained I tend to support historically bad teams and I can tell you that this idea of necessary pain or time fixing anything is BS, the only times I have seen any of the teams I support good was when someone(a head coach) highly competent was introduced. Across sports those clubs/franchises are the Canadiens, MHR, Waikato Chiefs, Timberwolves and Vikings. In my life time these teams have largely been a source of misery for their fans but sometimes when by some miracle something good happens, it happens due someone competent appearing and the effect are always immediate. The idea that it takes time is BS, that's not how sports works, things change very quickly, good and bad. Do not take competency and talent for granted.
PS: Because I'm an outsider I believe that 4 months is a short period of time to assess Amorim but that's only because I'm an outsider, if I was a sporting director for United, he would be in my office explaining precisely why he has struggled and how he intend to improve it without purchases, the quality of his answer will determine how I feel about his future. There is no scenario where I blindly give him time and money, not after these 4 months.
