I think one thing that needs to be called out here is the difference between "not right now" and "never". I suspect - purely from what I've seen in various places - that many people are being told the former and hearing the latter.
The big problem right now is that vaccines are still much more scarce than we'd like. And so if it's not hugely urgent that you get vaccinated right now because you're young, low risk, not in a critical job, not likely to expose anyone else who's high-risk... then yes, not giving you the vaccine so that that dose can go to someone else who needs it more is probably the right thing for society in general. This will change in due course as the more vulnerable get fully vaccinated and the overall objective changes from short-term harm reduction to long-term herd immunity. At that point anyone who wants vaccination will be able to get it, doctors will stop telling people to hold off, and the problems will change from "not enough vaccine doses" to overcoming vaccine hesitancy and hostility. That might not be too far off; it sounds like parts of the US are beginning to get there now, and we could be there in a couple of months if things go well with the vaccine rollout.
The other issue here is the fact that the clinical trials and approval process have been pushed through as fast as possible, and people fear the consequences of shortcuts that may have been taken. That isn't unreasonable, but hopefully it will subside with time; the process has been pretty good thus far and we are, after all, in the middle of a pandemic. The vaccines may not be perfect, but it's important to maintain perspective: a disease that's still killing thousands and thousands of people every day is way, way more frightening than a vaccine that has caused a small number of blood clots. If we want to wait for full clinical trials to be done as they are for most drugs... well, we could, but that will take *years*, not months, and the current intermittent lockdowns will continue throughout all that time. Is that really what we want?
Of course, there will always remain some folks who will never be convinced, whether because they're full-on conspiracy theorists like. The anti-vaxxers were around well before covid and I have no doubt that they won't go away. Hopefully they won't be successful enough to ensure we never come out of the situation we're in now... but it's entirely possible. They have managed to create a huge amount of fear around proven safe vaccines that have been around for decades, and a new vaccine for a new disease is much more fertile ground for them. This could turn out to be a real problem for society in general if it prevents us ever emerging from the pandemic, and if it becomes a serious threat then we will have to decide what to do about it.