lots of good points above
some additional blathering if it helps from someone on the post side:
the Godfather was made before the vast array of nuanced colour correction tools we have today were available or achievable (you had limited chemical/lighting based CC that could be done at the film printing stage of course) so a lot of creative choices had to be done in camera as it were (eg exposing indoor tungsten lighting on outdoor daylight film or vice versa, gelling lights, etc) - same with FX, as CG didn't exist back then
A LUT is essentially a colour transform that says for a variety of defined points in the colour gamut shift from original value A to new value B. They have limited resolution (ie number of defined points in a 3d matrix that covers R,G and B) so when applying a LUT there will be additional "interpolation" of how values map from A to B on any colour values that fall in between those defined points.
This is a long way of saying a LUT in general is an imperfect tool, though these days they can do a lot, and they can effectively encapsulate most end results of possible colour correction strategies, including making stuff warmer or cooler or otherwise tweaking 'white balance' in post. They are "dumb" however in that the transforms they describe are set in stone pretty much.
as other have pointed out the way a LUT works in practice to achieve a "look" thus relies on the source material (ie what you have shot on file) being consistent in terms of the changes it will make also being consistently predictable. The best way ensure that is to shoot consistently, and then do a prepass colour correction to make it even more consistent before any 'creative look' colour correction (inc via LUTS) are applied on top
OTOH this doesn't stop you also doing things similar to what they did on the Godfather in camera (eg by exposing tungsten at 6000k instead of 3200k say so everything looks 'warm') but on a digital sensor it will work slightly differently from film. If the image is 'baked' (ie you are saving to anything but a raw format) then the camera itself is already doing lots of subtle things to the image based on what it thinks (or rather the engineers who designed it think) is going to create the best image in the given limits of the format its saving to, and the sensor HW itself and those assumptions might not work well with the intended effect. You could say the same thing with film as well I guess - its just with Digital there's way more going on.
White balancing in post (inc using a LUT to do a similar transform) is different from white balancing in-camera because of that 'image baking' that happens in camera, which almost always throws out information in the process for one reason or another (usually to reduce file sizes or computation overhead in the camera) that may have consequences re achieving a look later on (eg a 'successful' white balance change)
interestingly if I look at that godfather clip you linked to I do rather think it feels quite odd colour wise out of context (yes I know its a classic!)- so again - there were analogous issues in film days also but far less room for maneuver as well