Physics professor here. As others have said, this is a really cute idea in physics that touches on some neat properties of the universe (like fundamentally identical particles, conservation of lepton number, and the effects of time reversal on particle properties), but:

1. It's more akin to philosophy than to science: this suggestion either has no testable consequences (or if it does, its predictions look obviously false: see below).

2. In all of our observations of the universe there seem to be many more electrons than anti-electrons, but this concept would seem to imply that the number should be exactly equal. You can try to get around that, but anything you do is a stretch.[1]

3. The formalism of quantum field theory naturally includes plenty of situations where electrons and anti-electrons form "closed loops" in time. (The classic example is a something like a photon giving rise to a virtual e-/e+ pair that immediately annihilates back into a photon, but they can get a lot more complicated.) Those closed loops would not be connected to the hypothesized "one electron" bouncing back and forth through all of time, so this idea would fail to explain why they also look identical to the rest.

So in my book, the beauty of this idea lies entirely in the fact that it could be suggested at all. It doesn't actually match reality, but it does give some striking intuition about how particle physics works.

[1] You could suggest that there is some other (unobservably distant?) region of the universe where antimatter dominates, but there's absolutely no evidence to support that suggestion. As another idea, the quote in the link suggests that "maybe the extra anti-electrons are hiding in the protons or something". But anything of the sort would seem to eliminate the whole point of saying "it's all one electron" (which always has the same properties, etc.).