Singularities and the Big Bang, Luminiferous Aether and Quantum Field Theory

In recent years I've been staying away from quantum physics because it doesn't make sense to me.  Is this a problem with me, or with quantum physics?

One post I read (Piekniewski) pointed out:

And above everything there is the singularity itself, a clear indication that something is wrong with this solution... My stance is that nature does not like singularities. Any time our equations suggest a singularity (particularly followed by paradoxes), we likely have the wrong equations.

And this makes sense.  The conclusion that black holes are singularities, and that the big bang resulted from a singularity are mathematically dubious.

In mathematics, a singularity is in general a point at which a given mathematical object is not defined, or a point of an exceptional set where it fails to be well-behaved in some particular way, such as differentiability. (Wikipedia)

So we have conventional wisdom positing a singularity, i.e. a point of undefined mathematics.  Yet, the whole basis for positing the singularity is mathematical -- there is no intuitive explanation of  the quantum physics of black holes and the big bang.

Much of quantum physics has been experimentally confirmed, but the parts regarding singularities in black holes and at the big bang can never be tested, much less confirmed.  My conclusion is that the problem is not me (or my unwillingness to tread the mathematically ardous road to singularities).  Rather, the conventional wisdom regarding black holes and the big bang is wrong.  I guess that there are mathematically sound explanations of these phenomena that do not require positing the magic of singularities.

With regard to quantum physics in general, there seems to be solid reason to believe in accept quantum field theory.  However, conceptually and intuitively, such fields  are similar to luminiferous aether, supposedly discredited in 1887.  I suppose there are some properties that were postulated for luminiferous aether that are different in quantum field theory, but this is not clear.  Force fields are everywhere and there are thresholds for realization as particles of energy or matter.  This is something that makes sense intuitively.  The conventional wisdom that we should accept mathematical singularities without intuitive support is less likely to be true or meaningful.