(US) Applying w/out hearing back from PIs? (Psychology)

I received advice from current PhD students in my field (Psychology) that basically said don't bother applying to a PhD program unless you connect with a PI in advance. So I've been sending out emails, but I haven't received any replies. Then I came across one PI's website that basically said (nicely) that he doesn't respond to emails, but he does read all applications, so prospective students should just apply if they're interested.

How common is this? Should I still apply to programs if I don't hear back from the PIs I'm interested in working with? Because at this rate I'll have zero applications if I wait for a response, and time is running out.