I think it's actually fairly hard to break organized infantry formations with archery alone.

Think about Thermopylae.

Even at Carrhae, what won the battle was that the Romans were demoralized because they had no way of hitting back, not because the HAs were killing a lot of people.

Even if archery did break the Romans, it took the Parthians ALL DAY to do so.
Well that's what the horse archers do. Kill and demoralize, and they are made to drag out a fight. It's not a weakness, it's part of their strategy. Darius couldn't beat the Scythians because they were perfectly willing to drag on his chasing them, demoralizing them without commiting to one big, quick battle.
As for Thermopylae, there wasn't room for a prolonged shower of arrows. I think that that is a bad example, do to extremely questionable tactics by Xerxes. A good general would have gone around, and not even bother fighting the Spartans.