[1] https://en.wikipedia.org/wiki/Nemesis_(hypothetical_star)
According to the linked article, it will take about fifteen billion years for the moon to escape Earth orbit entirely. I haven't checked, but I believe that's considerably longer than it is expected to take for the Sun to expand as a red giant and render Earth uninhabitable. Certainly more wondrous than either event would be for anything then living to remember having once had distant ancestors who called themselves "human"...
[1] https://www.google.com/amp/s/public.nrao.edu/ask/what-happen...
But the odds of such a collision I suspect are significantly lower even than of a similarly sized impactor striking Earth. Granted that the moon helps a great deal in sweeping space near Earth of objects that might otherwise enter our atmosphere, but something with the kind of kinetic energy it would take to produce these outcomes isn't going to spend long enough within the gravitational influence of the Earth-Moon system to be significantly affected before impact. It'd be like trying to swerve an ICBM warhead by blowing really hard on it, and I choose that simile with care - if this ever does happen, I think it might constitute a clear indication that someone somewhere in the universe really hates us!
The thing is that on the moon the craters remain visible for a much longer period of time while on earth they get covered up by plate shifts, erosion, ocean impacts, etc. So instead it might be a way to calculate how many asteroids likely have hit earth in general throughout history whether we have found the craters on earth or not. Then you could calculate based on the size of the craters on the moon which ones would have been earth catastrophic events (taking into account any diminishing impact energy due to our atmosphere). Which would tell us how many the moon saved the earth from. In terms of how many since human ancestors were evolving we would have to then date earth catastrophic craters on the moon. I'm not sure if this can be done through photographs or if we would have to get surface samples.
Then comparing that number to the fraction of earth covered/protected by the moon to the total area on earth we would have a decent idea of how many of these events happened one earth even if we haven't discovered the evidence for them on earth.