Yeah, I hate those "grumbling" cement slabs. Ya give them everything, and they still complain and grumble about things.
The crumbling you are experiencing, however, is different. Even a nice pleasant, congenial slab with a positive attitude can begin crumbling.
No fault of their own.
One, it could be the result of a poor finishing job. Often times right at a door, the finishers will make the mistake of working up extra slurry with sometimes too much water content to help "level out" the threshold area for a smooth, flat entryway. Not a good idea, as it makes the concrete weaker in that area. And later, it begins to flake off.
Two, the door installers may have had to do some chipping hammer adjustments to the threshold concrete. Not a great method, but sometimes done. This tends to make crumbling a little easier.
When it is so thin, patching is often ineffective. So, you can cut it deeper (down to about an inch at least) and go with a good patching compound suggested by your local concrete experts. Or you COULD try grey colored epoxy. It will also flake if applied too thin, but I have had some surprising success with it.