I couldn't imagine placing a tenant in one of my units without running their credit first. With that said, my properties are in A neighborhoods, and it affords me the opportunity to be extremely picky when placing my tenants. I'm extremely critical of credit scores when it comes to applications, and I've turned down countless potential tenants as a result. I've never had issues filling vacancies, and I've always ended up with tenants that were financially sound going into a lease.
I screen to the fullest to put me in the best position as a landlord. It's not a perfect science of course, and there are bumps in the road, but it avoids inheriting a known problem.
As for a property manger not running a credit report, that seems crazy! I would think that would be a big part of their provided service. I'd be most concerned about finding out that they don't run credit checks after having hired the PM. That seems like something that should have come up before enlisting their services.