I wonder if you would please justify that statement, specifically and only with respect to one of the things called the [Hawking-]Penrose singularity theorem? In particular, what was sufficiently unrigorous or alternatively missing from classical General Relativity that blocked a reasonable choice -- especially the early 1960s ones (example and commentary below) -- of such a theorem?
[Penrose 1965] Gravitational Collapse and Space-Time Singularities https://doi.org/10.1103/PhysRevLett.14.57 (2.5 pages)
(You can also find a copy in the usual place)
(I submit the reason it took until 1964 for a Penrose singularity theorem is explained by the first two lines of [Penrose 1965]: the surprising discovery that QSO 3C 273's highly extragalactic redshift z ~ 0.16 having been published in 1963 and QSO 3C 147's z ~ 0.55 following in 1964 strongly suggested SMBH activity. In other words the underlined statement in the fourth paragraph did not follow any sort of mathematical development (ADM, for instance) but rather was motivated by one of the most provocative observations of nature I can think of off the top of my head, up there with the (also 1964) discovery of the CMB.)
In a similar vein, I wouldn't expect that the singularity theorems necessarily hold for some of Einstein's early attempts at GR.
The problem with string theory is 21st century mathematics is still in its infancy. I have reasonable confidence that string theory is essentially correct only that the present maths and our understanding of QM (which needs improvement) is definitely not up to the task. Many topics like the Penrose singularity theorems in General relativity were impossible until Einstein's theory was formulated correctly with rigorous mathematics. IMHO this is the case with string theory. It simply won't work with the tools we have.
Another thing about String theory is that it very well could be a complete description of physics on anti-deSitter space. That would be progress but it still wouldn't be a unified field theory. We would need to find a more general theory which allows for deSitter space.
So there's my opinion. If anyone doubts mathematical beauty leads us to better things, plain old vanilla Classical mechanics is looking stronger than ever before and is still yielding interesting physics. Part of the reason for this is that the mathematical foundation of CM were still not understood properly until the mid 20th century. Give the other theories time to catch up. Mathematical beauty is the best guide we have. If strings turn out to be wrong then that's fine. Mathematical beauty is still the best guide we have. If we listen closely enough to what the equations are really saying then we'll find something better.