This seems to be of a type with one of the three common justifications offered for monarchy. 1. It's popular, 2. Tourism! and 3. Vague, abstract musings about legitimacy and consitutional roles.
The legitimacy of the British government doesn't come from the monarchy - it comes from the fact that it's a government elected by popular vote. As for what makes us a democracy - the fact that people think this is the most appropriate way to run a country in the modern world.
The only legitimacy that really matters is popular support. If the monarchy were done away with tomorrow, this would not suddenly mean that popular support for democratic government would evaporate along with the silly little quaint traditions. Maybe a written constitution would be appropriate, although given that absolutely nothing would change in the practical functioning of government, it doesn't seem wholly necessary.
I saw the movie with Helena Bonham Carter, Lady Jane (1986). Lady Jane Grey, "Queen for Nine Days", against "Mary, Bloody Mary". Two different factions, both vying for the throne. Who was the legitimate ruler? Mary won out, and that is the reason I assume to be behind England's perennial mistrust of Catholics and the Monty Python perennial "Nobody ever suspects the Spanish Inquisition!" Ultimately, the question has to always be who has the legitimate claim.
The question that actually matters here is who wins the armed struggle for the throne. Legitimacy is a vague, abstract concept, while the swords of the soldiers marching for you and the pikes of the peasants rising in your support are concrete, significant matters. If you can convince enough people to support you because of your marriage, or your bloodline, or the words of God, then you can win the throne. If you can convince enough people to support you because of backroom promises, a winning personality or promises of bread and circuses, then you also win the throne.