Nothing to do with fingers
People have come up with all kinds of clever ways of counting in various different bases using their fingers and/or other body parts, but there's no evidence that any base system was developed because of the ability to do so. Indeed, the fact that it's possible to count in so many different bases rather suggests no such preference even makes much sense. The best explanation for why different bases have been preferred at different times is exactly the same as why different languages, alphabets, and so on have also been used at different times - coincidence and habit. A language or numerical system or whatever evolves naturally, and then people keep using it because it's what they're used to, until it evolves into something else or gets pushed out by a new system for a variety of different reasons.
As for the article itself, the tablet is interesting but the comparison to trigonometry makes no sense at all. The whole point of trigonometric functions is that they are the ratios of the sides of triangles (OK, it gets a bit more complicated, but that's how they were first developed). In fact, the Babylonian method is clearly more primitive since they only address special cases, much like the Egyptians and others who also knew a bit about triangles long before Pythagoras*. The big deal with trigonometry is that you're no longer stuck with a few special cases and laborious tables, but can instead use general functions to handle any case you like.
* Pythagoras himself quite possibly knowing nothing about triangles at all, with no evidence linking him with mathematics at all until over five centuries after he died.