ding
It is said for example that the sum of all positive and negative energies of the universe are exactly 0. So when the universe started (all positive and negative energies united) it had the energy 0. In this case your "infinitely dense objects are impossible"-paradigma plays no role at all for example - totally independent whether it is right or wrong to say so.
And when we define masses as points in the spacetime - what we do - then every of this masses has an infinetelly dense gravitation in this center point (division by zero) - but this drives no one nervous because we never fall into the middle of the earth. And even if we could do so - because we are only gravitation and nothing else - then inside of the earth the masses would tear us in many directions and in the middle they would tear us into all directions.
But you are not able to know whether you will not fall into the middle of a black hole and that there is indeed all gravitation of the black hole (="infinitely dense"). What should stop you to fall into the middle of a black hole?