It's funny how people kind of pick and choose the singularities they are comfortable with. There are certainly plenty of GR theorists of note, such as Kip Thorne, who do take the predictions of GR seriously enough to think there really is a singularity in there. The story goes that what we regard from our external perspective as radius is converted by the extreme curvature into what is locally regarded as time, so then anything that experiences a forward march of time must reach the central singularity (or the ring singularity). In that picture, to say that some kind of exotic matter could resist gravity would be like asserting that it could stop time. Of course we don't really know what happens in there, but people like Kip Thorne do think it's a singularity.
What's more, the acceptance of singularities is not as rare as you might think-- people seem content with certain types of singularities. For example, many QED theorists are content to imagine that the electron really is a point particle, not just that it is small. Even more common is the idea that photons really have zero rest mass, which is also a type of singularity because then they have no rest frame. So we should at least be consistent-- we should reject all singularities of any stripe, or we should accept that any theory that works which includes singularities raises the possibility of real singularities. The hybrid approach where people pick and choose out of personal taste seems a bit disingenuous to me.