Where is the electrical panel located? Going outside the house may have been the path of least resistance (easier than routing through the basement). Could you route the ground cable through the basement joists and punching out to the original point of contact for grounding.
Would dropping it into the ground/earth and dig it down to prevent it being cut/damaged not work? After all it is acting as a ground for your panel.
I don’t really see a problem with painting it either as you are not painting the core of the cable anyway.
Something else to think of is if the water feed to your house is poly or some other material that does not conduct electricity. The water in the pipe is the only thing, supplying the ground to your home. Something to think about. Your ground rod is likely doing what it was intended too.
Could the grounding cable be taken to a grounding plate and dug in a couple of feet. See pic.