Kled rape jax in early, all u have to do is just not trading, u can just def from his dmg with e after 6, sheen and maybe phage u can already kill him, they key is to wait with your e to the moment when his lizard goes away, now u use e , he is slow, and can't really reload passive with his aa because u block it with your helicopter, he dies.
Vs fiora skill match up, if jax avoids riposte and use his e in good moment he wins, fiora overall has better early against him cus she can poke him. If jax get hitten by riposte, even without stun he is getting high as nerf, in that moment u have to go back or use e.
Trynd can't really kill u if u are using your E well, u don't have any chance early too vs him because he has crit an sustain, but after u get tabis, tiamat and maybe trinity or less, u just get bulky, u can duel him.
Hecarim also outsustain your dmg early, u just don't trade with him, after trinity u can just jump on him and smash the ground with him, if u didn't feed him, just farm, he pushes the lane fast so u can easy farm under tower.
Yes pantheon is less opressive than jayce, riven, first he is very easy to gank, and he just got early, if he don't stomp the lane, after 6 or higher lvl he will get rekt by jax. Just check highelo riven gameplay vs jax, and u will see what is the problem laning against riven as jax, can be adrian riven. Kennen is annoying, but can't really outdual jax in mid game. All what he is doing is just poke and running away if u are trying to engage, he can't one shot u, u can farm, there lies more about wave menagment skills.
Gp is annoying i must admit it, but still u outduel him if u really are not that much behind in creeps after first back for example. Jax with 2 items can kill gp with 3 or 4, I know from my experience because i once feeded gp as jax, and just when i had 2 items he had almost 4 and i won duel xd.
I wrote that they all are skill match ups for jax, except the ones i wrote below, they just make jax life miserable from the start, to the late game.
Yes the burden lies on the jax side to be a fucking sage of all champions and how to trade with them, and if u use that knowledge u can use him well. If someone pick jax because thinks he is op and can just go in on every AA reliant champion in every stage of game and win in every moment then he is a fool.