Then you haven't spent much time looking at actual textbooks on GR, since all of these issues are addressed there. (Not to mention in many peer-reviewed papers in the field.)

> if no matter appears to reach event horizons, how can they (event horizons) form in the first place (i.e. exist in our universe)?

This is the same fallacy as the fallacy that nothing can actually fall into a black hole because it looks like it takes forever from the outside. Oppenheimer and Snyder published a mathematical model way back in 1939 that shows how a black hole can form in a finite time from the gravitational collapse of a massive object, as seen by an observer falling inward on the surface of the object. The collapse appears to take forever as seen by a distant observer, but this is an optical illusion caused by the effect of spacetime curvature on the paths of light rays. This has been studied for decades and is thoroughly understood.

> While it takes forever for matter to fall in (again to outside observers), it takes finite time for black holes to evaporate via Hawking radiation.

Wrong. First, if matter is falling in, the hole is gaining mass, not losing it, so it will never evaporate even though, in principle, it is emitting Hawking radiation (in practice this radiation is many, many orders of magnitude too faint to detect for any black hole we can observe).

Second, if you include quantum effects, and therefore Hawking radiation in your model, you've changed the model, and it is no longer true that a distant observer will never see anyone falling into the hole. Instead, the distant observer will see light signals emitted by objects falling through the hole's horizon at the same time the distant observer sees the hole evaporate. (At least, that is the case in the simplest model, the one Hawking used when he first published his prediction of Hawking radiation. More complicated models have been developed since, and we won't know which, if any, of them is really correct until we have an experimentally confirmed theory of quantum gravity.)

> I'm not a physicist, but my speculation is that Black Holes don't really exist, they're just a limit of a process that approaches but doesn't converge to a singularity, which I think is unphysical.

Your speculation is uninformed and wrong. It is possible that an actual black hole, with an actual event horizon, is impossible when all of the laws of physics, including the laws of quantum gravity, are taken into account. This is an open area of research. But if it turns out that actual black holes cannot exist, it won't be for any of the reasons you give.