Physicist Proposes Alternative to Black Holes

At risk of sounding like those armchair physicists on the internet (oh wait, that's exactly how this post sounds!) it's "obvious" that a black hole cannot form.

The standard lay explanation of what an observer outside a black hole sees as an object falls into it is that gravitational time dilation freezes the object just at the event horizon. "[A]n object falling into a black hole appears to slow as it approaches the event horizon, taking an infinite time to reach it.[62]" (wikipedia citing Carroll 2004). For an object with just short of black hole mass, this time would instead be just short of infinite. So how does the black hole finish forming? (A similar problem seems to apply in the interior, where hawking radiation evaporates the black hole mass after essentially no time has passed for the observer falling inward)

Changing argumentative tactics to something totally nonphysical and unrelated to black holes except by sharing the term "singularity", this feels to me like a function such as f(x) = 1/x or f(x) = -ln x as you approach x=0. Yes, if you can get all the way to x=0 you're at a singularity. But if you're riding the curve on your bicycle, the distance you've got to ride along the curve is infinite, so you can never experience this supposed singularity after a finite time.

This armchair viewpoint is obviously flawed in some way, or real astrophysicists wouldn't be talking about actual black holes like they existed, so I'd love to hear an explanation that tackles this specific objection.