I like Tryndamere, but he isn't invincible.
If he falls behind, he can splitpush. But if HIS team doesn't know how to handle it, it can go just as bad. I played a game as Tryn and we got crushed early game. I was 2/3/1 and slightly ahead on CS coming out of laning vs an Irelia, not bad, but our bot lane fed a bunch of kills to the enemy and they really shut us down there. I was the only one to take any enemy towers, and we never got a dragon. The other 4 members didn't group up, didn't use towers for defense, got baited a lot, really didn't know what to do. And if I showed up, I got focused and killed in the teamfight because it wasn't really a teamfight. It looked like those movies where the bad guys throw 1 person at a time at the hero and he dismantles them each easily. When I tried to split, my team would also split up, and then the enemy would go as a group to each one of us and pick us off.
I had another game as Tryndamere in which I got ahead early thanks to an awesome early gank by Zac which gave my 4 kills and a tower. Screw splitpushing. I was in those teamfights every chance I could get. I finished 27/5/6, and spent most of my time in mid lane with occasional romps around the map to grab objectives or solo baron. They tried to split, but we stuck together and pushed to the end. They didn't really have much CC so they never could pin me down, and my team helped out a lot, unlike the other game where they had 4 different champs with good CC. (Morgana, Warwick, Irelia, Malzahar)
So, the game I won...stuck with my team. The game I lost, tried to splitpush. Of course it didn't help that I made a few dumb mistakes, too. So, yea, he was born to splitpush, but it only works if your team can handle it and the other team can't. And he can teamfight given the right compositions. He's not broken. You just don't like him because he's hard for you. Master Yi with Sated Devourer? Maybe broken. Runeglaive Ezreal? Possibly broken. Annoying Tryndamere? Just annoying.
Plus, if he kept dying after every objective he took, and you didn't use the 30 seconds to 1 minute plus to punish his team, then you were sorta doing something wrong there.