J.,
I imagine your son's doctor told you to wait because many children (especially boys) are not physically ready to potty train until after 3. They may not have the control, or they may not be able to tell when they need to use the bathroom. If your son is showing signs of readiness, then it is possible that he is physically ready. You won't know until you try.
I really only have my personal experience to draw on here, but my daughter is a good example of a child who showed interest in potty training but wasn't physically ready to train. At three, she understood what the potty was for and wanted to try to pee, but when she sat on the potty, nothing happened. Five minutes later, she would be wet. Rather than frustrate her and us, we let it go for a while. After a few months, something clicked, and she finally truly understood. It seemed like she trained in a couple of days. Now she has excellent control and stays dry through the night. However, without reminders throughout the day to use the bathroom, she will have three or four accidents. She rarely goes on her own. When we ask her why she didn't use the potty, she says, "Because I didn't know that I had to go." So even though she is wearing underwear and staying dry through the night, I don't feel like she is completely trained. Now my daughter is almost 3.5, and she is very independent. She doesn't want to be reminded to use the bathroom. In fact, she HATES it. We've built using the potty into our routine (pee when you get up, pee after breakfast, pee before snack, etc.), but this doesn't really feel like she is "completely potty trained" to me.
So I guess what it boils down to is what you consider potty training to be. Is it that your son is able to know on his own when to use the potty and stay dry throughout the day, or is it that your son understands what the potty is for and will use it when asked. For some children (like my daughter and my 5 year old son), the reminders feel like nagging and it turns into a battle of wills. But for others, possibly your son, since he is younger, reminders are fine.
I'm not sure if this is helpful, just another perspective. Good luck!