You're correct. You can't apply for the PGR document until you have the letter from the embassy. The time limit on the document is 30 days.
Also as a quick tip, my state (Baja California) has an online service where you can pay by credit card and they mail you birth certificates to wherever you want (certified mail) You guys might want to check on that and it might save you a trip or two.
The reason (my own opinion) why birth certificates have to be fairly new, is probably because old birth certificates in Mexico used to be handwritten, they looked horrible (think a doctor writing you a prescription!) As an example, I have one with me from 2009 and another one from 2012, and they do look a bit different.