This is a complex topic, but you ultimately seek to control displacement. At higher frequencies the displacement for a given input level will be lower. The methods of increasing damping you mention are largely velocity dependent, so will be more effective at higher frequencies (as Voicecoil said), so you still want stiffness to push those frequencies up.

Arguing stiffness vs. damping in a pure sense is meaningless, but as I said before, adding stiffness (and I really should also say mass) is easy, and should intuitively make sense in limiting system response. Trying to engineer in damping usually means allowing relative motion of elements of the system somehow, which undermines keeping it stiff.