The very idea of a singularity is an absurdity.
But you don’t have a black hole without one, do you?
So you’re dependent on infinities all the way through, it’s the very basis of the assertion that black holes exist.
Actually you don’t need a singularity to have a black hole. That’s just a mathematical inevitability based on our current knowledge of physics. A black hole is only the result of an object’s escape velocity exceeding the speed of light. And according to general relativity, it should take eternity for a singularity to form anyway, due to gravitational time dilation effects as the black hole collapses. We don’t really know what would happen at the point of singularity.
Singularities probably don’t actually exist. They are hypothetical constructs used for describing the math of what WOULD happen, if matter reached the center of a black hole. But matter never can go past the event horizon. This is known as “cosmic censorship.” Although in 1991, a couple of physicists suggested that at a small enough scale, the potential for a singularity might not be trapped inside an event horizon, suggesting that a real (or “naked,” because it’s uncensored and uncovered) singularity MIGHT exist. But this doesn’t mean that they do, or that they aren’t even nonsensical from a real-world point-of-view.