So I'ma ramble a bit here, but I' like to preface by saying that I agree that Sans is an incredibly interesting character and that this speculation here is very well thought out.

Anyways,

I was thinking about it and I think it's safe to say that "determination", among other things, has some play in characters being able to manipulate time. Whenever you approach a save point, the game makes a little comment about the character being determined by something nearby. It's implied that humans are the only known beings that have determination.

It's also demonstrated that not many monsters can really fathom what determination is. When they try to understand or make use of it, bad things tend to happen.

Frisk and the Fallen Human are unarguably the most powerful/determined beings we see in the game. A close second appears to be flowey, who can also manipulate time but is overpowered by Frisk's wishes. The only other characters that I can recall that are somehow aware of determination are Sans, Alphys, and Asgore (not counting asriel).

Your speculation implies that Sans is a former physicist that worked alongside someone (gaster?) as the previous royal scientist before Alphys on some other timeline. Something happened, in what appears to have been a mishap with an attempt at time travel which caused something really devastating to happen to Sans, and likely his colleague.

What comes to mind when considering this is the events that take place in the True Lab when Alphys attempts to utilize determination by infusing it into other monsters. The monsters affected appear to fuse together and become a completely different abomination entirely. They're still monsters, just really, really messed up ones with fused SOULS. The outlier of these experiments is Flowey. Flowey has the ability to manipulate time, but is now a SOUL-less, uncaring sociopath as a result.

My guess is that Sans and his colleague were also attempting to utilize determination in an attempt to travel across time, operating under Asgore as the royal scientists. Something went wrong which likely destroyed sans' colleague and caused sans to resign, but not before affecting sans in a way that allows him to use time/space travel to a certain extent. Does sans have a limited amount of determination as a result of his experiments? Is that why he bleeds red blood or has his unique eye and can apparently teleport around? Were they possibly experimenting with the Fallen Human's determination? In whatever case, I think that experiment might be the machine we see covered up in sans' shed.

You'll also noticed I mentioned Asgore as being aware of Determination as well. In the True Lab, it says in the journal entries that Asgore instructed Alphys to carry out the experiments with determination on other monsters. This leads me to believe that Asgore likely also instructed Sans (and his colleague) to use (the Fallen Human's?) determination to experiment with manipulating spacetime as well. Asgore's awareness of Determination and time travel is further evident by his fights with him at the conclusion of the neutral path; he keeps count of how many time's he's killed Frisk, just like Sans does on the Genocide route and Flowey does (sortof) on the pacifist route. Asgore is also the voice that speaks to you during a Game Over (unless sans is dunking you).

TL;DR I think San's excessive power and unique abilities might be the result of experimenting with determination on a previous timeline, which gave him limited control over time manipulation among other things.

It's also possible that I'm forgetting things that would negate my theories but whatever.