I think this thread is going in a good direction - that is, maybe he is ready to ride on his own and has had enough of being carried like a baby. Maybe it is his way of letting you know, even if he doesn't know himself.
My youngest daughter rode without training wheels once she had just turned three and I know a kid who was really competent at it (even did jumps) when he was still 2.
My best method (being the third kid I reckon she learned the easiest because we knew how to teach her!) was to start them on a 12" wheel bike with training wheels and back pedal brake. Start on a level surface with minimal rolling resistance (our wooden deck was perfect). All our kids did this once they had turned two and it taught them to pedal, turn and stop, (although I know teaching them to turn without leaning is a bit counterproductive, but at least the training wheels give them confidence and ability to go and stop and they love the freedom of getting around on the bike).
Then once they can do that put the seat down as low as it goes and take off both the training wheels and the pedals - if the cranks come off easily, then those can come off too. This transforms it to a kick along bike like the First bike, but the good thing about it is that it handles just like it will once the pedals have been put back on so they can progress without ever having to get used to a different bike. I agree with above posts that a slight downhill grassy slope is best. Other than that, just kicking it along is easy and within a week or two, my daughter was ready to put the pedals back on and she was away. During that year when she was 3 she moved up to a 16" wheel bike and before she was 5 she was doing some pretty hard core mtb on a 20" wheel Giant mtx250. Now at 7, she doesn't really slow us down much at all and happily rides stuff like the Scott24hr course at Stromlo.
My point is that the longer you carry them on the back of your bike or even a trail-along bike, the longer it will take before you can all truly mtb together properly. Also, getting them riding early will be great for their skills as they have a lot less fear than older kids. For our youngest, she is 7, nearly 8 and has been riding well on her own for almost 5 years while some of her school friends still have training wheels (and possibly frustrated parents who still can't take the family riding for a fun day out!)
Bottom line is that 3 years old is not too young.