The Speaker of the House does not have direct control over Capitol security or the National Guard. The Capitol Police are overseen by the Capitol Police Board, which includes the House and Senate Sergeants at Arms and the Architect of the Capitol not the Speaker.
The National Guard can only be deployed to the Capitol after a request from the Capitol Police Board and approval from the Department of Defense - which is part of the Executive Branch.
So yes, the President and the Executive Branch do have a role in how and when the Guard is deployed.
Which brings us to Christopher Miller, who was appointed Acting Secretary of Defense by Trump just days after the election. Miller was a yes man who placed restrictions that required high-level approval for the DC National Guard to respond with certain equipment or engage with crowds.
On January 6, this translated into a delayed response - despite urgent calls for help. Miller later testified that he wanted to avoid a military “overreaction” due to the optics, but many see those decisions as part of why the Capitol was left vulnerable during the attack.
If you don't like it - take it up with - 2 U.S. Code § 1901a and Article II, Section 2.
Additionally, the Insurrection Act of 1807 provides statutory authority for the President to deploy military forces domestically to suppress insurrections and enforce federal authority.