I don’t understand why this Anthony Alexander (progenitor of the North Carolina Alexander family) would be the son of William Alexander, 1st Earl of Stirling, 7th Baron of Menstrie
Sir William did have a son Hon. Sir Anthony Alexander, Joint Master of Works to the King who died without issue.
From https://landedfamilies.blogspot.com/2013/10/80-alexander-of-menstri...
Sir William Alexander (c.1577-1640), 1st bt. and 1st Earl of Stirling
He married, 1601, Janet (fl. 1649), daughter of Sir William Erskine and had issue:
(2) Hon. Sir Anthony Alexander (d. 1637), educated at Glasgow University; licenced to travel abroad, 1626-29; Master of Works in Scotland, 1629-37 (jointly with James Murray until the latter's death in 1634); knighted, 1635; married Elizabeth, daughter of Sir Henry Wardlaw, bt. of Pitreavie but died without issue, 17 September 1637;
I don’t think anything concrete is known about the father of the NC brothers to be honest.
There’s info in the profile for the 2nd Alexander at https://www.werelate.org/wiki/Person:Anthony_Alexander_%282%29