Infinite values (e.g. the density of a black hole, the size of the universe) and infinitesimals (e.g. continuous space-time) are assumed to be real, rather than inaccurate but useful approximations.
Not a physicist but I thought the Planck length specifically denies infinitesimality? And one of the proposed solutions to the black hole information problem is that due to local relativistic effects, they never actually reach singularity in finite time.
The Planck length isn't a minimum length or size, and doesn't have a whole lot of real significance other than as it results from a particular choice of units.
Assumed to be real by whom? Not all physicists believe the same thing.
Also why is this unfortunate? Why does it matter if they do or do not "believe" it? Are they able to make useful predictions with their models? Are they able to better understand physics? If so, why worry about their personal beliefs?
If we want to put physics on the faith table, I'm cool with whatever anyone wants to believe, and more power to them. But you can't be objectively right on the faith table - that's the price of admission.
If we want to be "true" and fully rational then we need to try to accurately represent our degree of knowledge about the world. Thus we shouldn't be making strong statements about things with an absence of evidence.
I wouldn't say assuming infinite density of black holes involves an absence of evidence. It is a hypothesis made in advance of evidence, and it leads to specific predictions that should be falsifiable, and in that sense it's considered scientifically sound.
As for implications that can't be falsified, for example ones that (to butcher Douglas Adams) "rather involve being on the other side of the event horizon," scientists can and do feel free to disregard those. Contradictory assumptions by scientists do not imply that one of them is "wrong" unless their theories imply contradictory observable phenomena. In which case there is probably a fruitful experiment to be done.