I would appreciate if somebody with a better understanding of GR could elaborate. I know there is no proper general local definition of gravitational energy but I always had difficulties on this aspect.

The answer to your question is subtle, and it's too late here for me to come up with a convincing explanation. Nonetheless, this claim

does deserve comment. There's no difficulty with defining energy at a point in general relativity, just as there's no difficulty in defining global energy; it's the notion of a quasi-local definition of energy which GR seems to lack, i.e., energy in an extended but finite region of spacetime.

Laszlo Szabados has many good papers on the ArXiv on this subject.

Yes, that seems more accurate even to me
Thanks for the comment

I am not too sure, here he deals with point masses on a background fixed spacetime. Intuitively, I would guess, if one does not fix the background, then point masses will be black holes in GR. Maybe that is what the author meant. But black holes are "allowed", at least several authors in the past have tried to describe fundamental particles as "sort of" black holes.

It is not that you can't have a point source for gravity (theoretically) but that with non-linear equations you can't "add" solutions. That is you cannot treat an extended mass as being a "bunch of point sources" as you could with Newton's theory.

Yes, he deals with a point mass as a black hole, and the point mass does perturb the background. However, reading Stingray's comments, there is no "source" here, since everything is a vacuum solution.