From a quantum perspective the Deutschian and similar models are honestly pretty compelling. They essentially require matching up the past and present in a consistent way that can remove paradoxes.

These make the most sense because it's entirely possible to write down spacetimes that contain "closed time like curves" (CTCs) ie. paths connecting past and present and you can then just let physics play out on these models (or more commonly using black box quantum circuits). The only consistent way to do it is to make sure the past and future side of such curves agree. It's not my area at all, just something colleagues of mine did, but from memory there are nice approaches using the path integral formalism that work really nicely in these scenarios.

All that's to say that I don't think time travel leads to anything changing, the past will have always agreed with whatever time travel happens in the future.

Having worked very briefly with the spacetimes that produce CTCs, I don't expect we'll be able to time travel because they usually violate the weak energy condition.

https://en.m.wikipedia.org/wiki/Quantum_mechanics_of_time_travel