Other research strongly backs the Covid vaccine's protection, but the flu half of this claim is left hanging.

Other research not linked in the post found that Covid vaccines strongly protect people from getting Covid. One study showed the vaccine works very well to prevent infection, though another looked at vaccinated people who still got Covid and found their T-cell responses were weaker after a booster, hinting that protection may depend on those responses. That T-cell study didn't directly measure how well vaccines prevent infection, so it's more of a mechanistic clue than a final answer. The big catch is that none of these studies tested flu vaccines at all, so they can't tell us whether the flu shot part of the claim holds up. No contradicting evidence turned up, but the evidence we have only covers the Covid side of the story.