There is a local cemetery with roads/paths that are open to the public and walkable – but the cemetery rules prevent running.
Is there a sensible way to mark these paths in OSM as “not runnable”? Or do I have to resign myself to walking these routes instead of running, to complete the city?
In case anyone is interested, I am referring to Mt. Auburn Cemetery, is in Cambridge and Belmont, Massachusetts.
When I did Mt Auburn, I would do a run and dedicate 1-2 miles of the run to walking in the cemetery at a brisk pace. It was a nice way to break it up. I may have also jogged for just a short distance at a time when I thought no one was looking. Maybe I did that. 
Similarly, this may have happened to me. I went early in the morning, to do the same approach you did … brisk walk and maybe jogging a little if I was certain that I was not bothering anyone, because there was no one in view.
So, I started jogging along at one point, when all of a sudden – from the apartment building across the street – an old woman yelled out, “THAT’S NOT FOR YOU!”
I felt kinda terrible at the time, but now it makes me laugh every time I am anywhere near there. 
That’s part of the <2% of Cambridge that I haven’t finished. I moved from the Boston area before their restricted pandemic hours and rules (only family members of the interred permitted) changed, but I had planned to run or BlueBike to/from the cemetery and walk those ‘streets’ since running is not permitted according to their website.