For a vaccination I'll call several weeks in advance. If I want an appointment before/after work I'll call well in advance, if I want something tomorrow they may say "11:30, take it out leave it", which isn't great for my work.
Don't trust whoever wrote what you read there. FUD scaremongers.
When we lived in Germany, we got TBE shots (area with a lot of ticks that carried it). IIRC we could also get an appointment in a few days.
(Though I think with vaccinations it's similar in the US.)
[1] Not eligible yet, so I pay 25 Euro to get one every year. Worth it for not getting the flu.
In this case GP probably knows that e.g. Italy has a life expectancy well above that of the US. Still, when he read that well-off people in the US have much better health coverage and it takes months to see a doctor in Italy, he believed it. How is it even possible to believe such a thing when you know the numbers?