What is Tryndamere Problems?

Split push is totally fun to play against.

My biggest complaint is RNGamere. Either he gets lucky and crits to win the trade or he doesn't crit and loses the trade. Either way, neither player feels very skillful about it.

This is a problem with Melee carries in General. They either do Too much damage and Bully Everyone, or not enough and can't lane. And unlike Marksmen, their lack of range makes positioning not very effective in the matchup.

Seems like the Best Auto Attacking Melees have a lot of In and out Abilities like Jax or Yas, But because of that, they arn't entirely AA reliant.

Not fun to go against, or have in your team

Heavily reliant on luck early game, like you said

Has the counter-intuitive mechanic of gaining power the lower his HP is

Counterplay-wise, there isn't much to him, asides from chain CC'ing him during an ult and/or kiting him, because you can't stop it from being cast, nor can you stop his W from removing a massive bit of your AD. Asides from turning around when he shouts, the rest is completely dependant on your comp/how fed he is.

Feast-or-famine like his mADC buddies

Ball of stats, mainly

Not OP, just flat out gimped so he can keep his ultimate with the same power as his signature demands

These are just from the top of my head.

Nothing like a champion who can become invulnerable for 5 seconds through hard CC.