Would some Brita (or any other related product) filter pitcher help at this, or it wouldn't catch some/most/all of the nasty stuff? (genuinely asking)
If you look at the list, even Brita doesn't claim that their pitcher filter the really nasty stuff. At most they "help to reduce". It looks like they are really designed only to remove the chlorine taste.
Their "Faucet Mount Filters" looks a bit more effective.