Jay wasn't interested in finding the Avatar. Sure, that's why he was called upon to join this team. But that's not his main goal.

He needed to find his birth parents. Jay found out recently, after being raised by earth benders, he was different. A fire bender that specialized in lightning bending.

Sure, finding the Avatar could help. He certainly doubts it's himself, seeing as he tried his whole life to earth bend and couldn't. But Jay's main goal is his parents.

For now, he rests.