My memory of Tomb Raider is a bit fuzzy (only played small portions of the various PS1 games), but I very much enjoyed the more deliberate acrobatic commands and leaps the game required of you. I don't see any reason why those couldn't be preserved in new Tomb Raider games, but I haven't played the new one, so I can't comment on that as to why it's not.
What I miss from games that do most of the climbing for you, and what I'm told is absent in the new Tomb Raider, are moments like this:
I love the series for its histocial backdrops. I somewhat understand why some people get tired of the series with all its games, but on the other hand i don't because they have a always have a unique historical settings, or beautifully created cities. Thats what i love, the gameplay may not change much but the setting is always changing. I mean we went from The Crusadest, to Renaissance Italy and Rome, to Constantinople, to the American Revolution and now the Golden Age of Piracy. I think thats a pretty diverse set of locations.