COVID Vaccine versus Long Term Symptoms

Writer Laurie Garrett tweeted:

....about new research reported in New Scientist that even mild COVID infections can bring long-term neural/cognitive symptoms.

And it didn't make sense to me - particularly the part about how these long term effects are prevented via vaccination. I asked the Slog's technical advisor:

If (as we know) the vaccinated can still experience mild infection, and mild infection can cause long term effects, then how can it be said that vaccine prevents long term effects? I don't see how all those things can be true.

Technical advisor answers:

First, New Scientist has been pretty bad in their coverage; not as bad as Lancet, but pretty awful nevertheless.

Vaccinees may be at risk of these effects, but, if so, it's very low, because while problem cases can theoretically "leak" through at each step, each step constitutes a reasonably effective filter, and the good news is that they're cumulative. So:

The Vaccinated Can Still Experience Mild Infection

Yes, but only a small fraction.

Mild Infection (among unvaccinated) Can Cause Long Term Effects

Yes, but only a small fraction.

Vaccine Prevents Long Term Effects

Yes, because exceptions are a small fraction of a small fraction.

Plus: the vaccine creates a third filter by also reducing the probability of infection turning into actual disease.

So because we’re concerned with a small fraction of a small fraction of a small fraction, vaccinees, in practical terms, need not worry about the prospect of lingering symptoms.