Does not the Queen herself actually have the power to control succession?
No. It's in the control of Parliament - laid down in Acts of Parliament. The most important of these is the Act of Settlement of 1701, but the Act of Union of 1800, and His Majesty's Declaration of Abdication Act 1936 (this last is now irrelevant, as Edward VIII/the Duke of Windsor died without heirs but it reflects the fact that even when a King abdicated, Parliament had to alter the succession because the King could not do so himself).
There’s been chatter for years (search engine: charles skipped in royal succession) that Charles would not be acceptable as King.
He’s an outspoken politically (alienating half of Parliament), was divorced, has a controversial business empire, and may have converted to Islam.
The Queen has informal influence, but only Parliament can make the decision.