Earlier quoted context omitted.
If someone says "if you don't fix this thing I will look for work elsewhere" that doesn't mean that you can say "I accept your resignation." If you want that person gone immediately, you have to fire them.
It's fuzzy. If someone in a heated dispute says, "I resign, effective some time in the next month", what do you do? Keep them on until you agree in a date? Even if they give a date like "Dec 31", do you let them keep all their access to sensitive company data, or you lock them out for security's sake and settle the legal and payroll paperwork separately?
If there’s no notice period (legal only in “at will” states, I believe) then it would seem reasonable to fire them immediately.