I'll have it for sure.
This is as much testing as any vaccines get; it's just been done in parallel. No waiting time for funding, for approval, for getting volunteers, for resources - everything just happens, at once. Most of the normal 10 years isn't because they are waiting and testing for 10 years. They are waiting for funding or other things instead - nothing is actually happening regarding the vaccine.
What would actually convince people? If you don't trust the scientists now, what could they do that would convince you? if you aren't a scientist, and you just have a feeling that things have been rushed, how will you ever decide that things are good enough? A feeling isn't really enough. You can only really decide by looking at the protocols and the data, and as most people don't have the background to analyse that properly, you have to trust the scientists who do. They don't want to do anything to harm people. they want the best risk/benefit ratio possible. They aren't part of the government.
Side effects will be apparent now. There aren't really going to be long term effects - that's not how vaccines work. There might be very rare side effects that will only be apparent once millions of people have been given it. That would happen with any vaccine, not because this one has been rushed. There is a very small risk of that, but that risk is the same with any vaccine, even those that have been tested for years - you only get that sort of data once you've tried it on millions of people (which shows how rare they are). This could happen after a year or 10 years, depending on the uptake of the vaccine. It really isn't something that is going to apparent just because of the passing of time.
Contrast that with known long-term effects of covid, which seems to be a very real risk. The risk of unknown side effects of a vaccine has to be smaller than that, because if they've not been found in the tens of thousands that have tried it so far, they will be rarer than that. If any unknown effects happen once it's tried in millions of people, they will be extremely rare - i.e., one in millions. That is rarer than the known effects of long covid.