My thinking on this was the exact opposite. Emergencies like this are precisely when you need these types of agencies the most, because you're talking about treatments that will be taken by a huge amount of people in a short timespan. Negative effects from treatment could end up as a catastrophe on their own. It's more important than ever to get it right.
Also when I tried seeing what people that actually develop vaccines had to say. It sounded like crash programs to produce a vaccine would take a year before there was enough capacity to mass vaccinate people. And then six months to vaccinate everyone. The required trials didn't slow anything down because they were done in parallel.
Also, some of the vaccines they developed for covid didn't work. A year ago people were somewhat dubious about mRNA vaccines because they'd never been tried before. And then previous vaccines attempts have also resulted in vaccines that make the infection worse. Or caused dangerous side effects.
I don't think with vaccines there is this 'one neat trick' that professionals have overlooked somehow.
And even six months into parallel speculative manufacturing and multiple vaccines approved or on track to be approved for emergency use, the US--in spite of vaccinating at one of the highest rates in the world--still needs to up its rate considerably to hit 50% of the population over the summer sometime. The idea that if the FDA had only washed its hands of the whole approval procedure thing in March, things would be fine now is just a complete fantasy.