By turn 3, the Infernos start kicking in so he is gonna win with or without a weapon.
With a few exceptions, the turn Jaraxxus is played, all he can do is attack for 3, and Infernos won't kick in until turn 3.
Consider Alexstrasza, she also heals you up to 15 and is an 8/8 that kicks in on turn 2. You also have the choice to bring the opponent down to 15. Jaraxxus is probably no better than Alexstrasza.
There are situations where Jaraxxus is really good, but that can be said for all 9-drops. Think Ysera, which is rarely not good on turn 9.