I just had a watch of the "first cave" video - it looks pretty much spot on compared to the original!
I'll give this a shot once it's progressed a little. I'm still waiting for the "Black Mesa" remake to complete - I must have bought that ages ago now .
Have been playing the demo and... it's annoying as hell
Lots of falling to death after misjudging jumps and leaps and playing the same part over and over and over again. No manual save, just checkpoints (in the demo, anyway).
I do remember not being mad keen on the first 3 Tomb Raider games for this very reason, too finickity and fiddly for my liking. I do suspect though that for lots of players that's probably the appeal.
Good graphics, fluid gameplay, works well with Xbox 360 controller but not for me.