Not necessarily. A black hole doesn't have any stronger gravity than an ordinary object of the same mass. And matter falling into the hole would radiate strongly (the paper calls this "accretion luminosity"), which would help to maintain an equilibrium with the rest of the star. That's the kind of model the paper is studying.

How long such an equilibrium can last is a different question. The paper only briefly comments on this when it says that the time scale of the numerical simulations they did is of the same order as the hydrodynamic timescale of the Sun. That means, roughly, the time it would take the Sun to collapse to a white dwarf if fusion reactions in its core stopped, which is, I believe, tens of millions of years. So a star with a black hole at the center would not have the same lifetime as an ordinary main sequence star with similar mass, but it would have a long enough lifetime that would could not conclusively rule out that at least some stars we see have black holes at their centers.