If this unbounded state were asymptotic then it wouldn't be unphysical, but a freely falling particle reaches the singularity in finite time. A freely falling observer crossing the event horizon could in principle perform this measurement as they approached the singularity, and general relativity could no longer describe their measurements.

Okay — you’re still not explaining what actually breaks, just repeating they could measure over and over.

Please say what you specifically believe is unphysical about the situation — what trajectory reaches the singularity in finite time and why specifically is that unphysical?

My understanding is that you have a cusp singularity that is actually an infinite spike, ie, distance to the singularity is unbounded; that is, no matter how small a circle/sphere around the singularity, you have an infinite diameter. And so you will need to be much more explicit about where the problem lies.