My personal opinion only:
I use oil, not as a finish, but a suppliment for leather after tooling. For me (I am not chemist and I do not play one on TV either) I find that tooling dries out the leather causing it to become a bit stiff when it dries from the stamping. Now, there are a lot of variables to the above statement. I based this on my own experiences.
Now, I use two types of leather sources depending on what I have on hand or need last minute. I am at the end of my Hermann Oak stash, I purchased 5 double shoulders from Siegal of CA and in a pinch I have purchased some of Tandy's imported leather. For my Hermann Oak, it is a toss up when I tool on whether I use oil. If it a big project that will take a lot of carving and stamping, I will probably apply oil after tooling. My Siegal leather is also a toss up on the application of oil depending on the status of the leather part. The Tandy Leather is almost definately in need of oil after tooling. I could commit homicide due to the stiffness after tooling. On some of Tandy's leather (the one that has a 8 o'clock shadow of skin on the flesh side) I will definately have to apply oil to both sides. I've thought about making a swatting paddle out of a piece due to the stiffness of the leather.
As for which oil I use. I have found (again, my opinion) that the neatfoot oil will darken my leather a bit, but more than a bit on Tandy leather or any other blonde colored veg-tan. The EVOO will not cause a darker shade after application. So...........If I am using Tandy leather or my Seigel leather, I may very well use neatfoot since I know that it will turn it a darker shade of brown. My Hermann Oak, not a problem, stick EVOO on it and move on.
I would highly recommend that you test out pieces of leather that you have on hand to see how things work out for you!
Good luck