Does Science Truly Represent Reality?

Yes, there is actually It is a hang-over from SR that gives the incorrect impression that time is relative. The Universe itself provides a standard of time - that measured by a 'comoving' observer. We are such an observer, ignoring our non-relativistic motion caused by our Galactic orbit, and Galactic local peculiar velocity. It is certainly possible to measure a smaller time since the BB, simply by ensuring a constant relativistic motion (or equivalently hanging out on the fringe of a black hole since the BB), but you cannot get a longer time. The 13.7 billion years is the maximum time it is possible to measure since the BB. This makes sense in relativity as the longest distance between two points in space-time is a straight line - i.e. a non-accelerating observer.

By Galactic orbit I simply mean the orbit of the Sun around the Galaxy centre - which is far too slow to be of interest. 'motion' created by the expansion of the Universe is not motion at all - as you say, not the same thing as velocity. Exactly.