I do think T's screen first time clients, but usually after they have a phone conversation with them. For example, they might decide that you are not a good fit or that your presenting problem is not in their area of expertise, or that they have "too many" of "x" type of client after speaking with you.

That being said, are you leaving really long messages with alot of detail? If so, perhaps shorten the message to the basics and ask for a call-back.

Also, similarly to sittingwatersedge comment, if a T can't return a call, even to say "I am overbooked" and provide references, do you really want that T?