Short answer: no.
That particular manufacturer's vaccine saw a doubling in the narcolepsy rate, amongst those who took it, it went up to 1:50000 as opposed to 1:100000 who would get diagnosed with it in a normal year. In a clinical trial - statistically speaking at that level, you might not see any cases of it at all, and you won't have enough cases to see a pattern.
Longer answer: in countries where swine flu spread rapidly, the rate of narcolepsy cases also went up and at a similar rate.
It's important to note that the reason why that story is discussed is that it is because it's a rarity. There have been hundreds of new vaccine formulations since then, and billions of vaccinations. Serious vaccine side-effects are rare, that's why the same incidents are talked about a lot.
It's like a lot of things in life unfortunately - there's a risk analysis to be done. We know the lack of a vaccine has already killed about 1:1000 of the population in the UK, and that's despite us living with some pretty miserable restrictions. I don't believe any approved vaccine will kill or cause longterm harm to 1:1000 of the people taking it, I don't even believe it will do it to 1:10000.
Beyond that (as we start to talk about 1:100000 or 1 in a million) we won't know for sure until more people take it. We do know what will happen if people keep getting covid though - a lot of deaths, a massive overload on the health service, and a large number of people with long-lasting illness.