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Astra-Zeneca and Johnson & Johnson are based on a different methodology to the other vaccines, and AZ and J&J are the only two that I've seen any reported harm, in the form of a characteristic type of blood clots (which have sometimes killed) in a "small" number of people. They've figured out a possible mechanism and they suspect one of the additives - EDTA I think - is associated with it. So personally I wouldn't want either of those types of vaccine. I doubt the risk is great - about 1 in a million - but why take it when there's a choice? There's been no similar news of any such problem with the other types, and given that the news about AZ and J&J got out, I can't see why it wouldn't also have got out if the other types also had side effects. That's why I have rather more trust in Moderna and Pfizer. It doesn't mean I'm 100% certain they'll never hurt anybody, but weighed against the benefits it seems the best bet, and like I said earlier, I got away with it.
I don't quite agree the UK gov's advice to have AZ. They did seem to be playing down its risks when they advised older people not to refuse that one, and I can see why they fudged the advice a little. From the perspective of the NHS, the gov and the company, it's good if people accept AZ. From the perspective of the individual, it's not quite that good. To give credit where it's due, they decided not to give young people AZ, because they said the balance of risks and benefits is rather different for them, being less likely to die of Covid.
I think you're right that they did rush the vaccines out, which suggests they may have cut a few safety corners. In fact that's why I decided to wait until a few million people had been jabbed before I got mine. They were my willing guinea pigs.