It's not a pop culture term at all. It's a precisely defined technical term in GR. Everything I said about it is taken from that precise technical definition.
> A singularity is just where physical quantities go to infinity.
Correct. (Although this itself is something of a pop science version of the actual technical definition. But it will do for this discussion.)
> There are many singularities at the event horizon.
Wrong. All physical quantities are finite at the event horizon. Your counterexamples are based on misconceptions which have been well understood and corrected in the GR literature for decades.
> as you fall into a black hole and approach the event horizon the time of an observer a distance away begins to speed up and approaches infinity
Wrong. If you are free-falling into a black hole, light from the rest of the universe is redshifted, not blueshifted.
If you are hovering at a constant altitude above a black hole's horizon, light from the rest of the universe is blueshifted; but the blueshift is finite for any altitude above the horizon. It is impossible to hover at the horizon.
> the universe comes to an end before you actually reach the event horizon.
Wrong.
> Many unphysical things like that happen.
Wrong.
> the energy density of space-time approaches the plank energy
Wrong. The black hole is a vacuum solution: the stress-energy tensor is zero everywhere.
> this is a region where general relativity and quantum field theory do not agree with each other. So we know the equations are non-sense in this region for sure.
Wrong. The area at which GR and QM create compatibility problems is within a Planck time of the singularity, not the horizon.
As I said above, all of these misconceptions have been known of and corrected in the GR literature for decades.