If I wanted to cripple a societies science, introducing religion is a pretty good way to do it. (But no probably not)

What about aliens tho

Very doubtful.

Time travel is science fiction, with the emphasis on "fiction". There is one vital factor that most time travel stories leave out. Our planet and solar system are moving through the galaxy at approximately 10,000 miles per hour. Out galaxy is also moving through the universe at incredible speeds, so if you were to travel though time, you would most likely end up in empty space, because tiem travel woudl rquire factoring every gravitational influence since the big bang to determine the exact spot in space to coincide with the time you are traveling to.

Add in the multi-verse theory (I wont' take time to explain this one, but there is plenty on the internet to explain it), and it is virtually impossible to ever create any form of tiem travel that would affect the tine line you originated on. If you did manage to affect your own time line, then you have the problems of timeline paradoxes to overcome.

So, the probability of your worries is so inconsequential that numerically it would makes the speed of light look like the smallest of conceivable numbers.

That's basically the premise of Card's Pastwatch book: humans on a dying Earth observe that their past was manipulated by humans from an alternative future, realize that they can also manipulate their past to create an alternative alternative future, then do so under the guise of being "divine emissaries" from prehistoric deities.

There are some "oddities" with regard to health in the texts of the Judeo-Christian faiths such as the Kosher rules which helped to minimize diseases like Trichinosis among their populaces and there's that NT passage that basically describes the foundations of CPR. More than likely these rules/ideas were based on repeated observations/practice but there's also a small chance that there was some Wibbly-Wobbly, Timey-Wimey shenanigans going down and those were the only bits that survived in any recognizable form.

Yep, could be.

If the anal probe doesn't fit, to religion, you must not submit.

Evidence would be extremely hard to find indeed.
As I don't see how we can remember a past that never happened i.e. if something changed the past today, tomorrow we would only know of the new past...

I propose a tool to help identify the possibility of a consciousness that can travel in time and space and maybe fiddling with our past. So, assuming math is invariant between timelines and Benford's law describes the distributions many naturally occurring properties; we can look for natural distributions that do not follow Benford's law. As Example, an area of the world has some lakes with a totally random distribution of the surface areas, or economic data from right before a war is wonky, maybe that data could be reviewed for explanations.

I fully agree it is a funky far out idea.

There is a very good book on this subject called "Time Travel in Einstein's Universe" which will give you a very good handle on whether or not any beings are altering our past. Understanding the physics of possible time travel is somewhat difficult but even the everyday person will get an idea of the possibilities.

"Deny the possibility (...of water having been changed into wine in the Bible)? Certainly not. I would not deny any such possibility. But I would, of course, not spend a moment on it unless there was some evidence for it." - Carl Sagan

Perhaps a simpler explanation might be aliens having violated their version of the prime directive in our past. But then again. if the alien ship hiding at a Sol - Jupiter lagrange point 50 years from now observed us blowing ourselves up, perhaps they might have gone back in time to try to instill some morality in us by means of religion. Or not.