Not really. The only thing they do better against is Skirmishers (still lose) and Huskarls.
Low HP targets with good armor.
Even 3 base pierce armor is a difficult thing for them to deal with on anything heavier, like Eagles, Paladin, or Cavalier. Regular CA do a lot of damage, enough to bypass most peirce armor; at least, a lot better than Archers.
The thing that keeps them from getting outright slaughtered in these cases is the 0 frame delay. It lets them survive, and still requires micro. let’s not forget that micro requires attention, attention that could be put into your economy or elsewhere. HCA have a very, very good damage output. Even regular CA do once they get upgrades.