But which black hole singularity?
A black hole actually has two singularities, one at the center and one at the event horizon. Both singularities show up in common parameterizations of the Schwarzschild solution. Each could have been troublesome for Einstein.
The singularity at the center is based on the idea that all the mass collapses to a single point, so that the point geometrically blows up to something of infinite diameter. Physically, the idea is that if the mass is sufficiently concentrated, the gravity force will overwhelm all other forces, including the Pauli exclusion force. Then nothing can stop all the mass disappearing into the hole in spacetime.
The first modern solution of general relativity that would characterize a black hole was found by Karl Schwarzschild in 1916, although its interpretation as a region of space from which nothing can escape was first published by David Finkelstein in 1958. Black holes were long considered a mathematical curiosity; it was during the 1960s that theoretical work showed they were a generic prediction of general relativity. The discovery of neutron stars in the late 1960s sparked interest in gravitationally collapsed compact objects as a possible astrophysical reality.I don't get this. What did they think that a black hole was, before 1958?
This singularity is indeed a troublesome concept, but it is not clear that it has any physical significance. We don't know that gravity will really overwhelm all other forces. That belief is based on an extrapolation that can never be tested. Nothing inside the black hole can have any causal effect on anything outside the black hole.
The other singularity is at the event horizon. This was not well understood until decades after Schwarzschild. Now it is common for textbook to say that it is a removable singularity, or maybe not a real singularity, because someone falling into the black hole may not notice anything strange at the event horizon. I say "may not", because some argue that someone would see a firewall.
The event horizon does appear to be a singularity in some coordinate systems, and therefore to some observers. Someone watching the black hole might think it very strange that a falling object seems to take an infinite amount of time to cross the event horizon. That infinity is a singularity.
You don't really need to believe in singularities to study black hole physics. The singularities are not observable.