> They aren't real in the physical sense.
In the spirit of Oppenheim's CQ (classical gravity, quantum matter) work which is discussed in the fine article at the top, I'll say that your first sentence is a bit too strong. Curvature singularities (in the Kretschmann (or other curvature scalar) divergence sense) aren't ruled out by astronomical observation. Indeed, practically no observational data sheds any light on the question. What is easily ruled out is Schwarzschild black holes (known black hole candidates all have significant angular momentum) and Kerr black holes (Kerr & Schwarzschild are eternal without beginning while our universe appears to have a finite age; they exist in an energy-free and non-fluctuating vacuum rather than in a local region full of at least CMB photons but also gas and dust and nearby stars; and they exist in a larger volume filled with stars, black holes in their host galaxy, galaxies with other black holes, and so on).
At the following link is Roy Kerr giving a 2016 invited talk where he points out that despite the success of the metric that carries his name, especially for round spinning bodies (planets, stars) and even the then-current evidence favouring Kerr as a good description of large black holes (evidence since then has also been supportive), nobody should feel comfortable about the Kerr interior solution because there will be matter inside an astrophysical BH that is for practical purposes Kerr for outside observers. https://youtu.be/nypav68tq8Q?t=2884> (around the 48 minute mark).
This is not that General Relativity is wrong, merely that GR itself depends on a solution to the Einstein Field Equations (EFEs) for a given spacetime, and black hole spacetimes commonly known by non-specialists (including physicists who aren't relativists) have features which are unphysical and which actually matter. There are lots of somewhat recondite solutions to the Einstein Field Equations (EFEs) which look like Kerr or Schwarzschild black holes to families of observers, but which have very different geometrical structure (these model black holes might have formed a finite time in the past by collapse of matter, for instance, or they might couple to a more realistic expanding spacetime with gravitational radiation from distant souces sloshing around).
Known black holes (rather, things that in telescopes etc are for all practical purposes black holes) are in such a complicated environment by comparison that we simply do not have an exact solution for the Einstein Field Equations for any of them. We are stuck with approximations, and those approximations often don't even solve the EFEs (even numerically) but rather some cut-down version.
The Raychaudhuri focusing theorem and similar results make it pretty clear that singularities form rather generically in fairly generic curved spacetimes equipped with matter. Penrose has raised comparable arguments. Our actual spacetime is not really generic on the whole, so we look at small pieces at a time and hope we are right in our guesses about how we can assemble multiple small pieces into an improved approximation of our universe. Those small pieces are calculated to be riddled with singularities for reasons related to Raychaudhuri, but that's an artifact of the construction of the small pieces as solutions of the EFEs, how we truncate those solutions, and/or how we stitch them together (e.g. Darmois-Israel).
Maybe what singularities arising in commonly-used black hole solutions of the Einstein Field Equations tell us is that our understanding of matter is off. Indeed, the whole article at the top is about a programme to study how quantum matter influences the gravitational field ("back-reaction"). For all we know, real matter that remains outside a black hole and does things like inverse Compton scattering (as well as matter that plunges inward) blocks the formation of a singularity inside. Part of the motivation for quantum gravity (CQ being a flavour thereof) is being able to extract observables that can answer that question.
> Physicists do not believe ...
I'm pretty sure many physicists hope that nature doesn't produce singularities mostly because that makes it harder to ask how parts of the universe evolve (solutions to the EFEs become infinite at a singularity, but we can sorta work around the supposed singularity with Bowen-York punctures, dynamical excisions, adaptive mesh refinement and other techniques). I'm also pretty sure that anyone with a background that supports the formation of such a belief (or its opposite, or some third choice) knows that today we just don't know how to prove much (even in principle) about the interior of black hole candidate objects in our sky.
Conversely, I do not believe there are singularities in those astronomically-observed objects, but that's because there isn't much evidence one way or another. In practice it really doesn't matter because the interior doesn't really matter much in practice, and if singularities are somehow observed we will cope with them. As to "somehow observed", quantum gravity phenomenology researchers will tell you that we would be lucky to catch the leading order quantum correction to GR with present-and-near-future observational ability, while theorists will then tell you that the singularity-or-not answer probably depends on the next-to-the-next-to-the-leading order. Consequently we have practically no way to distinguish among quantum gravity theories (including CQ) which are known to reproduce the successes of General Relativity in the neighbourhood immediately around our own planet.
(Indeed, the gravitational field of Earth barely needs General Relativity, let alone quantum corrections to that, and there are lots of things we don't know about the deep interior of our own planet. As far as anyone can tell using a post-Newtonian formalism one only needs the leading order (1PN) to fully describe all present measurements of gravitation around us. Earth's gravity essentially a barely-perturbed Kerr (exterior) metric (see e.g. Soffel & Frutos 2016). And that's only thousands of metres to thousands of kilometres away. So not knowing about the deep interior of things at minimum thousands of light-years away doesn't really disturb me, even as we go up to 3PN+ (see e.g. Clifford Will's 2011 "On the unreasonable effectiveness of the post-Newtonian approximation") based on our long-distance observation of the glowing and/or opaque stuff outside of them).