I agree with the vocabulary videos. Mine likes the Meet The...(Shapes,Colors,Letters,ect) and has learned his colors in under 2 weeks. You can contact ECI for early intervention and they offer speech therapy for children under 3. My son is very interested in puzzles,cars,computers...anything mechanical and analytical. His speech is delayed but not delayed enough to be covered by ECI( the child must have at least a 6 mo delay to be eligible). So I am trying the same methods you are(naming everything,making him ask for things,etc). Depending on what your son's strenghths are, he could just be processing the information differently like mine does, in which speech comes in last. If so, he will probably be very good at numbers and such when he gets older. Every child(and person) is different and we all input and output at different levels. I do not agree that there is a "norm". Just b/c something is common, doesn't mean its normal and that your son is "behind" just b/c he's not up to common standards. If he is at least trying, that is a very good sign and he will eventually catch up. Afterall, he has only been alive for 700 days. So no, 3 is not too late to start worrying. Too late to start worring is school age,5. Many children have a noteable increase in speech from age 2.5 - 3, so that is probably why your DR told you to wait.
But there is no harm in contacting ECI to see if your child qualifies and getting the speech therapy that is offered if it will make you feel better.