How did the time loop even start? *Spoilers*

It is a causality loop; the contradiction and problem of the grandfather paradox.

That loop is never-ending and should not have been broken as they did using a single timeline to reset everything.

Temporal loop series/films use a single reset timeline and/or multiple alternate timelines.
This series used both.

That's my complaint with it - similar to the original Terminator, the time loop is interesting but it could never have started.

As long as the loop isn't broken, I don't think there is a problem with Elias being born, then going back to be his own great great great grandfather, because he is still being born.

Unlike in, for example, Star Trek IV, where Kirk is given a pair of glasses in the future, sells them in the past, and then he jokes that they'll be given to him again in the future again. In this case, at no point in time were they actually ever made (I realize it probably wasn't the same pair).

But once the loop is broken, that's where the paradox happens. They go back to stop Elias, but then there is no reason to go back to stop him, so they don't, but then he's not stopped, so they need to go back to stop him...and so on and so on.