You've pointed out everything that makes Tryndamere a good laner. However, he has very little to contribute to teamfights, and this is the core of why he isn't very competitive. In the current meta the solo is expected to shut down key targets, mess up team composition, and/or tank for the team. Tryndamere can't really accomplish any of these goals unless he's far ahead of others.
Tanky Tryndamere is weak because he lacks the base ability damage that other solos have (Singed, Irelia, Darius, Jayce, Olaf) and he also lacks CC (his shout CC isn't reliable), so he isn't able to easily nuke key targets and he can't mess up team composition. The exception is when you're against a very heavy AD team, then his shout shines. However, the range on it is limited so Tryndamere tends to suffer lethal damage when performing the shout in teamfights or he simply never gets in range to snag everyone. His shout can rarely be used as a clutch play, especially because the slow component relies on your opponents facing away from you. Even Zileas has flagged this ability as having design issues that make it unsatisfying and relatively weak.
Full AD Tryndamere is also weak. Yes, his ult keeps him alive, but he is still vulnerable to CC and he has a hard time closing on a target and killing once his spin is down. A single exhaust shuts him down until he is killed. Compare this suicide style Tryndamere to suicide Karthus, who during Death Defied:
- Is invulnerable for 7 seconds
- Is immune to exhaust and cc
- Deals constant AoE damage without mana constraint
- Has a guaranteed global damage ultimate
Althought Karthus can't move during during Death Defied, it's very easy to position him in a clutch spot with flash during a teamfight.
Ask yourself, can suicide Tryndamere ever have presence like Karthus does in teamfights? I don't think so. Ezreal may have got the penta, but that wouldn't have happened without Karthus' presence.
Also, Tryndamere's only damage ability (the spin) is almost always used to gap close instead of doing damage, so he does not hurt as much as other full AD champions who have so many abilities that scale on their high AD. Full AD Darius, Olaf, Jax (AD/AP), Irelia, and Riven can easily outperform Tryndamere in teamfights.
If you think you can get ahead in lane and use that to your advantage late game, that is also not a reliable strategy. Yeah, there are some solos that he can duel, but he is incredibly easy to counter (Darius, Jayce, Olaf, Malphite, among others) and it's easy to zone him from his farm because he lacks early game threat. It's very situational to end up against a team that Tryndamere could both solo against and contribute in teamfights.
As for jungling, junglers tend to get less farm and experience than other lanes, and Tryndamere does poorly without farm. His ganks are decent because he has a gap closer and CC that works naturally well in ganks, but he simply doesn't perform better than other junglers in any way. Compare Tryndamere against Lee Sin who:
- Has better ganks
- Clears jungle faster
- Sustains in jungle better
- Has great CC and base ability damage
- Messes up team composition, even though he falls off late game
- Most importantly, can function on little farm
Lee Sin outperforms him in every aspect, so there's no advantage in picking Tryndamere. If Lee Sin is banned there are still better all-round junglers than Tryndamere including Udyr and Dr. Mundo among others.
TL;DR: There is nothing that Tryndamere does better than any other solo in teamfights. His solo ability is also less than stellar because he can be countered by a large portion of common solo picks. He is also outperformed in every aspect by junglers like Lee Sin, so there's really no benefit to picking Tryndamere.