I don't know what the NF rulebook says, but the NCAA rule is:
Rule 3-1
Art. 2. Each team may continue to play with fewer than five players when all other squad members are not eligible
or able to play. (Emphasis added.)
This seems to address many of the questions raised here. In the original case, one girl wasn't able to play because of sickness, and another wasn't able to play because of state rules. No problem.
Maybe the NF will see this thread and revise the language in their Rule 3 to match the NCAA language.