Okay, before I start, I always like seeing new Jax posts. Furthermore, I fully acknowledge I have a lot less points on the Grandmaster than you, but I'm also a main, so let's talk this out. Finally, I happen to disagree with you, but please don't ignore me just because of that.
I understand why you feel that his ultimate is out of place. I agree, at least, thematically. However, gameplay wise, the ability makes sense.
Jax's gameplay identity revolves around several facets. Let's go through them.
Hybrid Damage Output: Jax is specifically designed to be a melee hypercarry regardless of whether he builds AD or AP. This is reflected in his kit: while he doesn't have great AD scaling (restricted to his Q, E, and ultimate), he can hit the AS ceiling with only two items, some runes, and his passive; meanwhile, you aren't punished for building AP on an attack speed champion thanks to his Leap Strike, Empower, and Ultimate passive and active. Buffing either of these fields is a very dangerous affair, and why buff him anyway when he's in a good spot?
Dueling Strength: The price Jax must pay for his unholy damage output is the fact that he will still have to pentakill the enemy team one person at a time. Jax has only one AOE ability, so his main contribution in team fights is his ability to survive long enough to duel the enemy ADC or Mage to death. He accomplishes this with the combo of his ultimate's active and the E dodge/stun, then using his Q to jump to safety.
Counterplay is a requirement for every good champion design, though, so Jax needs a window of time where he isn't invincible, and that window is when his ultimate is on cooldown.
Glass Cannon viability: Jax is the Grandmaster at Arms. You know that. You have to love him for it. As such, he shouldn't be building tank items. Jax uses weapons, and then (in order to cater to his gameplay as someone who gets into a fight and STAYS) he gets defensive bonuses for them. That was one of the core identities for his theme when he was just released- this champion is meant to run into a fight, go HAM, deal over nine trillion damage, and get the f*ck out before he dies.
Thematically, it doesn't make sense for him to have sustain. His battle output is more like an upside-down parabola, high burst, sucky mid fight, and a destroyer of worlds when you end the battle. Sustain would warp his power windows to make him virtually unkillable by allowing him to survive long enough to kill everything in the late portions of the fight, which would be over powered, which would mean our purple friend gets nerfed, or worse, reworked, and then he'd be banned every game.
I don't think the problem is his kit. I think that ANY problem Jax might have is with the mediocre item choices he is faced with.