Generally speaking, the longer a fight goes on, the better it is for Tahm: his ult's passive lets him hit like a truck for building survivability, his E massively improves that survivability, and the uptime on his Q stun, provided you can consistently hit a marked target with it, is kind of bananas. His W also has some BONKERS burst, especially if he winds up with any magic penetration or magic resist shred. Tahm's utility is situational, especially compared to Braum, but strong in it's niche, and backed by high damage and high survivability.
With that in mind, what can Tahm do...
1 : persuade frontliners to leave your team in peace or suffer the wrath of Tahm's damage and CC: again, Tahm's passive is 'if you stay near me when you have a fish over your head, I mess you up royally'.
2 : keep a chase going. Tahm's Q has a short cooldown and a strong slow. Unless someone is particularly fast or has access to good escape spells, Tahm can generally catch up to anyone provided they've gotten in Q range at least once.
3 : ulting into fights. One of the best ways to turn a teamfight is to change the terms of that fight. Tahm's ult straight up adds Tahm and potentially another friend to a fight on the other end of the map. If your team is outnumbered in a fight you ult into, the enemy team either has to end the fight NOW (which is hard because they get less warning) or reposition quickly to avoid having to deal with an unexpected catfish.
4 : get a slowmoving melee character into a fight. A 30% movespeed increase when moving towards danger is not something to scoff at. Got a Volibear? Garen? Darius? Morde? Trundle? Galio? Eat them and march up to a fight you want to have happen. Unless you build straight damage, you're more durable than any of those champs, and with a 30% boost for several seconds, you're likely faster than they are too. It's not quite as good as having a Kalista throw them into a fight, but it's more flexible and less stupid.
5 : GTFO. 15% movespeed buff is pretty great for getting you and someone else out of a bad spot. Even something as simple as say, repositioning your artillery-esque hypercarry to a better spot, or just keeping them safe while spells go flying, is a HUGE boon. Also, if you and your passenger are pursued by a small group, a Q slow can get you some serious distance.
6 : Waddle into the enemy's backline and wreck them. You get a huge portion of your HP as damage to everything you do, and backliners generally don't build MR. Either you get there and kill them (which is good), they disengage (which isn't ideal, but still better than them attacking your squishy squishy buddies), or they try to blow you up instead of your team since you're an immediate threat and a target of opportunity (which is ideal because you get so much mileage out of survivability it's kind of bananas).
7 : temporarily remove a problem from the fight. 3 seconds of being in a 4v5 (or 4v4.5, depending on if you're in a position post enemy champ devour to still do anything useful) is a goddamn ETERNITY, especially since once whoever you remove is returned, they're returned stunned and missing a big ass chunk of HP. And this can work on all sorts of people. ADC getting out of hand? Tank locking your whole team down? Morde, Darius, or Skarner ruining your squishies and seeming unstoppable? Stop them with stomach acid. It'll give your team time to reposition and take out other targets, so that when the problem comes back, it can be dealt with in better terms.
Full disclosure : I'm a bit biased. I like Tahm Kench, a LOT. He's fun, he's charming, and he feels really impactful. And I'll admit, his teamfight isn't the BEST: it takes too much time for him to get going in a teamfight, and he's better and small scale catchouts and ambushes. Still, he's hardly useless. Just requires a bit more finesse than most people in his weight class.