Probably whether the English king claimed the throne of France depended on exactly how much they wanted to piss off the French at that particular moment in history. At least until the French chopped their King’s head off, then after that I think the English kings lost interest in applying for that job.
The Hundred Years War was fought because the King of England claimed he had a better right to the French throne than the King of France had. That war ended in 1453 but the kings of England continued to claim the title of King of France for more than 300 years after that.