@Robin4 You seem to be advising the host on how to solve the problem, when it was in fact the guest who asked for advice.
Considering that the guest has already requested a frying pan, and the host has already made a decision on that, the choices available to the guest are:
a) double down on the insistence that the host provide the item at his own expense, risking an escalation of conflict that would probably poison the well for the rest of the stay
b) press the spurious case with Airbnb that the absence of a frying pan constitutes an unusable amenity, as a basis for ending the booking early and going to all the trouble of finding new accommodation and relocating
or
c) just buy a bloody frying pan and be done with it.
As I've already weighed in, (c) would be my own choice, if I intended to complete the stay. There's a time and a place for righteous indignation, but you have to ask yourself if that's the hill you want to die on, considering how the problem could be solved so much more easily without ever having to ascend a hill in the first place.