But comments above state that the vaccine doesn't effect spread. If that's true, then vaccinating everyone would not address any of those points.

It hasn't been proven to affect spread (because that wasn't tested for). But it's generally assumed it does to at least a degree.

It's also much harder to test spread. You'd have to do repeated follow-up of all participants with frequent testing. When you have 40,000 participants, that's a significant concern. Trials like these are called "large, simple trials" since at that scale it's impractical to have a complicated study design at that scale. In addition to the pure expense, it's difficult to comply consistently with the protocol if it's complicated and you need to implement it at dozens or hundreds of sites. And recruiting people becomes harder and harder as you need additional follow-up, and attrition is higher (and likely biased: you're more likely to take your test if you think you might be sick).

So it's entirely reasonable to just study symptomatic cases in these trials.