Even ignoring all confounding factors it should still be possible to detect an effect via regression. It's like if you try to price S&P off just one of its constituent stocks; you're still going to observe some correlation even when only looking at a single large constituent.

The results of the other article I linked, https://link.springer.com/article/10.1007/s10654-021-00808-7, showing that vaccination does not reduce viral load, supports this notion.