Onceapilot :
Of course wx avoidance is not an emergency! But you do not seem to understand how the system works : ATC does not force you to go through it . ATC might refuse the diversion for various reasons and because of external factors :e.g military area, prohibited airspace even conflict zone (e.g Ukraine or Turkey , etc..) You then have the choice to divert to another destination , go back or possibly go through.
Just like Fog at your destination: Landing at intended planned destination is not mandatory under all circumstances. CBs are included in that.