Cannot like this post enough, Foxy earlier mentioned the same thing.
Jax isn't about staggering pressure like Tanya and Kung Lao where it is so scary to hit a button. He aims to use his slightly plus frames to either poke your counterpoke and get high frame ADV, (D1 and D3 are 15 and 16 respectively of the top of my head) and when he gets you scared to poke open you up with his overhead low. He can blow up most armor with 123 however all the options from this string put him in negative for his read. (Not downplaying the use of the nuts delay of the 3 in this string to hit people with Gotcha, RocketRC and in Wrestler the fantastic tick throw which works great against character with no armor)
For those who only know Jax in HW you'll understand how the rocket confirms are great for hitting conversions in the neutral (Used to be much more valuable when his combos were dumb) Against slower reversals he is also able to make use of a 10 frame mid to frame trap and cancel to make it safe/bait reversals.
But what if my opponent doesn't have any real armor I need to worry about? Let's take D'vorah for example the worst we eat is EX Ovipositor strike. Suddenly Wreslter has a huge advantage because every time this character counterpokes you he can then tick throw you when you do read him. He does 11 you respect his option of 123 and block and get thrown for 14% and this character is still all over you.
Let's say you have no good way of claiming space or have a weaker ground game that Jax, you might have good aerial specials/normals? Well now anywhere on the screen you have to deal with Pumped Up's IAGP, Plasma (Delayed or not) better Armored AA and a reversal that does 37% midscreen and 42% or 44% I think if you manage to reach the corner. So you've taken Jax's whole neutral and allowing him to zone and play defensive against characters that have trouble making space.
Went off on one a bit but Jax requires a lot of work and I'm still finding ways to best play each MU as each variation offers him the advantages you just need to flesh them out enough to see it.
TLDR: People sleep on Jax because he needs quite a bit of work to learn.
You hit the nail on the head with regards to Jax's pressure. Nothing ridiculous about it, it can be neutral ducked and there are gaps to exploit for those who know the matchup (I believe this is perfectly fair, by the way), but he has a lot of very practical strings that are very slightly plus on block. You really have to think and maintain an awareness of your opponent and his or her capabilities in order to keep pressure going consistently, but if you put the thought in and learn the matchups, the pressure is there. You just have to mix things up and play creatively, you can't go braindead and mash the same inputs out over and expect to have success the way you can with some other characters.
I actually think one factor causing people to shy away from playing him is because being +1 or +2 on block doesn't mean a damn thing online. When most people pick up a new character, they play it online before going to locals, and I think people internalize the difficulties maintaining any sort of pressure against good opponents online as their own failing, rather than the fault of the game, particularly if they have no problems applying pressure with other pressure heavy characters online. Somehow, with online, +1 or +2 moves become punishable, while -3 and -4 moves jail. It makes no sense to me.
I suppose part of the issue with his midscreens as well was not so much that it was difficult (it is one of the tougher BnB's in the game, but other characters have tough BnB's as well and people play them just fine), as much as that it was inconsistent, prior to the patch/hotfix (which was a godsend). My hand/eye coordination is far from S Tier, so if I can get his midscreens down consistently, anybody can.
I think people just have certain sub/semi-conscious expectations of how a character should play, and when that character doesn't meet those expectations, people assume that it simply isn't the character for them, rather than adapting to the situation at hand.
Another factor is the lack of high level representation he receives. Tyrant and Jupiter are the only big name guys repping Jax as a main right now, and Tyrant hasn't been placing as well as one would expect, and I believe Jupiter does not show up to a lot of tournaments (?) (although when he does show, he finishes top 8). As such, the character isn't as visible as others, and people don't have a consistent source or reference for higher level tech. Tyrant is the last guy you want to copy, as I think it is unrealistic for most people to expect themselves to be able to execute at his level, which is fucking unreal (super human even) even among professional players. You don't need to be able to execute at his level to win (most of the guys winning majors don't), although the expectation that you do can be discouraging.
Anyways, moving on to Jax tech, I gave up on instant-air ground pounds, as air ground pound is an abysmal -44 on whiff. Meanwhile, regular ground pound is a very good -21, and it low profiles most of the games mid projectiles, including Reptile's Forceballs and HW Jax's rockets. Air GP doesn't low profile at all. I basically only use Air GP now to remind people it exists, in the hope of baiting them into attempting a punish every time I jump backwards. I seriously think Air GP needs a buff, -44 is ridiculous for a move that does 6% and leads to nothing.
The regular ground pound is still godlike, however, particularly in conjunction with his energy wave. If I am just beginning some sets or a match against a top level player, they start off the game ducking my energy waves and jumping my ground pounds fairly consistently (as they both have relatively slow, reactable startups), but I notice that over time the mixup begins to wear down on them, and they begin messing up and getting caught more and more often, until eventually I can zone them out for a round or two. Obviously, I'm talking about players who are much, much better than me, so I still get bodied, but the ability to steal a round or even the occasional game based off of it against some of the better players on the scene hints towards its viability.
Plus, I have begun to notice that everybody I play begins to think their controller is messed up, and I have attributed it to the + frames from my energy waves (this is offline only, by the way. Energy Wave does not seem to jail consistently online). A belief that one's controller is not accepting inputs properly leads to clenching and other bad habits, which leads to more difficulties inputting. It was a pretty nice Eureka moment when I realized that that was the case.
So, yeah, there are a lot of reasons to play Jax. Perhaps the best uppercut in the game, 3 viable variations, decent and unique zoning potential with Pumped Up, lots of high-execution riff-raff in HW, and a viable grappler with lots of tic throw options in Wrestler, just to name a few of his perks. He has something to suit almost all types of players.