The notion of "centre of the black hole" is misleading. It treats the spacetime in and around the black hole as if it were normal flat spacetime, with no gravitational distortion. In fact the spacetime is highly curved. So curved, in fact, that "space" behaves more like "time". Instead of "where is the singularity" you should ask "when is the singularity."
The answer to this question is always "in the future". There are no singularities in our past. This is significant, since when we say "there is a pen on the table" we mean that the pen was there a few nano-seconds ago, because of the time light takes to reach us. Everything that we say "exists" actually exists in our past. Singularities are a "when" that have a "before" but don't have an "after". Nothing can have a singularity in its past, because there is no future after the singularity.
So in this sense, singularities don't exist. We can't study them because they are always in the future (due to the extreme bending of spacetime). If we fall into a black hole we will (rapidly) reach the singularity but there is no way to report back about it, since the singularity is surrounded by an event horizon. Some of us may (if we are unlucky) fall into a black hole. For these people the singularity is in their future. But it is not possible for a singularity to be observed, as it can never be in the past of any timeline.
So a black hole behaves "as if" all the mass was concentrated in one point. But don't think of the singularity as a "thing" in the centre of the black hole, but think of it as a "when", and always remember "this is weird, and analogies and metaphor are never a complete explanation".