why does the royal family play an important role in british festivals?

Answer 1

The power of the monarchy in England is largely symbolic today, and the Queen is the "mother" of the country, in symbolic terms. Festivals are rituals that create unity between people, in part by reminding them of the traditions that make them "a people."

The Queen's presence reinforces the present unity and the connection of the present with the past.

Answer 2

Because the Queen in Head of State and and as such takes the lead role in national festivals. If we didn't have a monarchy, then the President or other head of state would do exactly the same.