Update: It is one thing to suppose that the singularity is a mathematical artifact, and another to reject black hole formation entirely. In this regard, the article seems vague about exactly what Einstein and Rosen were saying in their 1939 paper. It seems to me to be saying that, having concluded that collapse would not stop at the Schwartzchild radius and that would lead to a singularity, we should conclude that collapse to the Schwartzchild radius itself is not possible.
The article linked to this point is (IMHO) well worth reading, and seems to support this view of Einstein’s position.
With regard to contemporary attitudes to verification, we know that Eddington went to Principe to measure the magnitude of gravity’s effect on light specifically because it would be seen (and was) as strong confirmation of the theory.
But then after Oppenheimer did the math in 1930 and showed that collapse into a black hole was within the theory of relativity would you still taken the same position?
Showing that a black hole was possible within the theory could just be a problem with the theory itself. Not necessarily an indication that they must exist.
From the Universe Today (at https://www.universetoday.com/122715/what-are-white-holes/ ):
"Black holes are places in the Universe where matter and energy are compacted so densely together that their escape velocity is greater than the speed of light. […] Fully describing a black hole requires a lot of fancy math, but these are real objects in our Universe. […] So then what’s a white hole?
"White holes are created when astrophysicists mathematically explore the environment around black holes, but pretend there’s no mass within the event horizon. What happens when you have a black hole singularity with no mass? […]
"Now if white holes did exist, which they probably don’t, they would behave like reverse black holes – just like the math predicts. Instead of pulling material inward, a white hole would blast material out into space like some kind of white chocolate fountain. […] One of the other implications of white hole math, is that they only theoretically exist as long as there isn’t a single speck of matter within the event horizon. As soon as single atom of hydrogen drifted into the region, the whole thing would collapse."
And from Space.com (at https://www.space.com/white-holes.html ):
"White holes are theoretical cosmic regions that function in the opposite way to black holes. Just as nothing can escape a black hole, nothing can enter a white hole. […]
"To a spaceship crew watching from afar, a white hole looks exactly like a black hole. It has mass. It might spin. A ring of dust and gas could gather around the event horizon — the bubble boundary separating the object from the rest of the universe. But if they kept watching, the crew might witness an event impossible for a black hole — a belch.
"Physicists describe a white hole as a black hole's "time reversal," a video of a black hole played backwards, much as a bouncing ball is the time-reversal of a falling ball. While a black hole's event horizon is a sphere of no return, a white hole's event horizon is a boundary of no admission — space-time's most exclusive club. No spacecraft will ever reach the region's edge. Objects inside a white hole can leave and interact with the outside world, but since nothing can get in, the interior is cut off from the universe's past: No outside event will ever affect the inside."
I would imagine that black holes show the limit of modern physics, there are still things to learn, and models of physics will probably be adjusted in the future.
I guess there are physicists who are able to predict how a black hole works and appears, and how it can be explained, but I haven't read about it.