There are several pieces of evidence that support the claim that Shakespeare did write to please the British Monarchy. For example, many of his plays are set in England, and they often feature characters from the royal family. Additionally, Shakespeare sometimes wrote plays that were specifically commissioned by the royal court. For example, he wrote "A Midsummer Night's Dream" for the wedding of Queen Elizabeth I in 1595.
However, there are also other pieces of evidence that suggest that Shakespeare did not write to please the British Monarchy. For example, some of his plays contain criticism of the monarchy. For example, in "King Lear," Shakespeare portrays King Lear as a foolish and tyrannical ruler. Additionally, Shakespeare sometimes wrote plays that were not well-received by the royal court. For example, his play "The Merry Wives of Windsor" was reportedly not well-liked by Queen Elizabeth I.
Ultimately, there is no clear answer to the question of whether or not Shakespeare wrote to please the British Monarchy. There is evidence to support both sides of the argument.