One reason that Mary was executed was because she had a strong claim to the throne because of Elizabeth being illegitimate to Catholics. For example, because of Henry divorcing his first wife before having Elizabeth against the wishes of the pope Catholics thought she was not really part of the royal family. This meant that due to there being no divorces on Mary’s side of the family, she was part of the royal family, therefore Catholics thought she had a stronger claim to the throne. However, this isn’t the most important reason as Elizabeth also had arguments to say that she should be on the throne
A more important reason she was executed was from her involvement in the Babington plot. The Babington plot was a plan to assassinate queen Elizabeth and put Mary in the throne. Mary was thought to be involved in this because a major part of this plot was to put her on the throne as well as there being a coded letter sent to her. This meant that Elizabeth had reasons to think that Mary was a focus of the rebels and therefore a threat to her being on the throne. Though this isn’t the biggest reason because there was no actual way to tell if Mary was the one directing this plan.
However, the most important reason she was executed was because of the privy council. The privy council were on Elizabeth’s side and wanted Mary dead. They also thought that as long as Mary was alive, she would be seen as someone better for the Catholics. This meant that when Elizabeth was about to sign Marys death warrant, they went and finished the job as quickly as they could. This was obviously the most major reason for Marys execution because without the council who knows if Elizabeth would have signed the warrant.
I personally think that the final point is the most important as without the privy council I doubt Elizabeth would have actually chosen to sign the paper and with that Mary would still be alive, just locked up like she already was.