Tilt, the point is that if we are going to discuss another tradition, it's important to explain the language of that tradition in the context in which it was intended. By focusing on the perceived "sectrarian claim" behind the term Hinayana, you are providing an incomplete picture of the other tradition that you are discussing.
As I wrote earlier, in any form of Buddhist practice, the approach we take only is beneficial to the extent that it inclines toward liberation. So I can understand that in a certain context, the term "Hinayana" could be useful to some people as part of a teaching about how to pay attention to one's own practice. And in fact that is how I understand the term "Hinayana," namely, not as part of a teaching aimed at judging others, but rather as part of a teaching aimed at helping one see for oneself.
I am merely suggesting, in the context of this thread, that the underlying teachings of various Buddhist traditions share important elements in common insofar as they offer a path of practice that can incline toward liberation. In my opinion, that's worth remembering. It's easy to get distracted by these "sectarian claims," as you put it.