Actually Tomb Raider: Anniversary is a very good remake of one of the absolutely best ever classic games of all time, the original Tomb Raider.
It features modern FPS-like controls and flawlessly working 3rd person camera, very nimble platforming, the classic 90s feeling of being isolated in another world, uncompromising difficulty, and at times awe inspiring level design, all the while respecting the original and its mechanics. This is why you have oldschool puzzles, traps and key finding, and it's just doing its job as a remake.
The grappling hook is an absolutely awesome addition to the game. It creates entirely new challenges for the player and adds variation to old ones. And this is what Tomb Raider is about, actually: brutal, murderous challenges, acrobatics and traps. It was too hard? It is supposed to be hard. TRA will absolutely make you want to smash your monitor at times. In this, it is true to the originals, which is its damn job as a remake.
Lara moves slowly? Perhaps, but it's not really an FPS, and the enemies are very fast, especially lions, panthers and the like, and they are trying to grab and tackle you. Yes, lions run faster than Lara. This is simply part of the challenge. Once you get the controls down, combat is actually very fun. You do have to use the grapple in some boss fights, this is a staple carried over from Legend.
http://www.youtube.com/watch?v=zSX_8zdu_zM
The over-the-shoulder free-aiming mode isn't for combat, it is for puzzles, hence why you can't move. There is that puzzle where you have to shoot lights on and off to make constellations, and the shootable ropes that make the statue fall down in Croft Manor or the bridges that reveal secret grapple points when demolished.
Not looking like Crysis? Well, TRA is largely an indoor game, since it's a remake of a 1996 title. Tomb Raiders are generally indoor games. It is rubbish that Legend looks better. Underworld looks better, though it's noticeably still the same engine.
TRA is indeed the hardest of the "new" TR titles; in this way, it's just an uncompromising remake of the old ones. If failing to do a grapple run 20 times makes you hate it so much you never come back, then TR isn't for you. It's a common trait of the franchise, and it's why hardcore fans love it. Why would you want an easy platformer. The multiple wallrun thing in the great pyramid level is indeed fucking hard, yes. And the timers are generally short in the game. But it's totally beatable without cheating just by using the game's standard controls and Lara's possible moves to their full extent. People are doing it for sport on youtube, so it must be doable. It's also the second to last level, so it should be OK to be difficult.
http://www.youtube.com/watch?v=mXj7hMmjUNE
I don't know what you did with the T-rex checkpoint - the checkpoints in this game always worked brilliantly for me. Probably bad luck.
You didn't mention the awesome secrets, which at times are much, much harder to get than some of the grapple-wallruns and are actually unique archaeological artifacts.
As for the traps - they're simple but effective. Simple elements like crushers and rotating blades create the challenge, you can't get more hardcore than that. The grapple adds an additional timing element.
http://www.youtube.com/watch?v=a2tgFOIAo2Y
I guess it's the same as many things - you like it, or you don't. It's not a bad game at all though; as a remake, it is just perfect. If the game it's remaking wasn't up your alley in the first place, you can't really blame the remake.
Cursing at the monitor is expected and socially acceptable when playing Tomb Raider. Early on, you could impale Lara or kill her off in a myriad of ways to vent your frustration, sadly this has been restricted and the most painful deaths were removed due to political correctness I guess :-)