Do crews look for "Dr." on a passenger manifest to identify medical doctors?
You are correct that it is done by PA rather than by reference to the manifest, at least at my carrier in my type of aircraft. As you surmise, the "Dr" notation has too many opportunities for both false positive AND false negative results to be where you want to start when time is of the essence.

Besides, if you need medical help at 40,000', you're probably much better off in the hands of a good E.R. nurse or paramedic than with a radiologist who hasn't done hands-on emergency care since his residency. 

Most airlines use a service that's reachd by a phone patch to an E.R. doctor who specializes in evaluating these sorts of situations (and recommending, divert now or continue on), so any medical professional on the aircraft is working under his direction. He is also familiar with the contents of a medical kit carried on the aircraft just for these situations, so he can say "have the (nurse/paramedic/doctor/etc) administer ___ out of the medical kit" already knowing what is &amp; isn't available.

Interestingly, a veterinarian can also administer the same medications the other medical professionals can in this sort of situation!