There's an obvious change in tone from Underworld that is not just a result of a bigger budget and better graphics.
Yes, there is, firstly, in terms of pure graphic violence, I'd say it's about equal in violence shown, with the exception of the visual difference inherent in much better graphics, but the tonal difference is despite Lara being historically butchered in awful ways, in this game, she's butchering people in awful ways. That is a very notable difference to her, but I don't think it is to the game.
Games are designed in iteration. Until now, CD have just been trying to stay true to the original game design focuses, and polish the experience, make the move set organic, have the game look better, have the progression across the game more logical, etc. They'd never got around to actually designing a wholly new game and addressing the decade long issues with the series. It was a chance to build the series from scratch, and when doing that, they designed a combat system that is a lot more visceral. Lara never had a chance to be really violent, because it was always her, auto-aiming, jumping around like an idiot. It's not that her character believed leaping around while firing dual pistols with arms outstretched was a good idea, her character wasn't aware of that action at all, because it was purely the game design, and wasn't ever an aspect of the storytelling. You were just meant to ignore the fact at one minute she looks completely moronic, and the next she's trading one liners with some minigun welding Italian sociopath.
Contrary to what they said on the podcast, I don't think Crystal Dynamics took over until after Angel of Darkness came out and flopped.
That's true yeah, they're probably confusing that Core had two teams at the time, TR:TLR was in development at the time time as AoD, and there was a staff switch up once TLR finished.