No, it doesn't. The single electron could still be acted on by various operators in the course of its twisty path through spacetime, which could make it appear in various different ways. Which operators might be acting is then deduced from the different ways electrons appear.

Your hypothesis amounts to the claim that the only operator that is required to act on the single electron, to explain all its different appearances, is the spin reversal operator. My point is that that hypothesis only accounts for the appearance of "electrons" (more precisely, different views of the single electron) with opposite spins and the same charge; it does not account for the appearance of "electrons" (different views of the single electron) with opposite charges, because no unitary operator can change the charge, and the spin reversal operator is unitary.

If this operator actually existed in reality such that a single electron's spin can be flipped by applying it to an actual physical electron?

Sure it does; there are many different kinds of experimental setups that can realize this operator. One of the simplest is an inhomogeneous magnetic field. The "operator" language is just a way of linking the math describing electrons with the physical setup that does things to them.

If this operator does exist, then one could not say that there was a single electron on a path, unless all electrons in the universe changed spin direction instantly after the application such an operator.

No, this is not correct. The operator can be localized, so it only operates on one "electron" (i.e., one particular segment on the single electron's entire worldline) at a time.

In fact, regardless of which operator acts on the electron, it has to be localized for the hypothesis that there is only one electron to make sense. The whole point of the hypothesis is that the single electron follows a highly twisted path through spacetime, so that it crosses any single spacelike slice of spacetime multiple times, with each crossing potentially having a different appearance (different spin, etc.). That means whatever operator acts on the electron to make its path through spacetime twist has to be localized--otherwise all the appearances of electrons everywhere in the universe at a given instant of time would be exactly the same, which they obviously aren't.