Yes, the marriage date makes it impossible for Bridget Higham to be the mother of Thomas French III (and the grandmother of his offspring that have birth dates begining in 1590).
The 1560 marriage date has no source and appears to be an estimate (?). I've updated the marriage to 1597. Richardson cites (1) Muskett, Suffolk Manorial Fams. 1 (1900), p 310-314 (2) NEHGR 108 (1954), p 172-178. Harvey Visitation of Suffolk 1561 2 (H.S.P. n.s. (3) (1984): 391-399.
I see your question regarding both Thomas French and Thomas Burrough having been married to an Elizabeth (Unknown), based on the lineage you linked and the French Family Association.
The French Family assocition does show a mother for Thomas (who christened ca. 1551-d. 1613, French-266) as Elizabeth (Unknown), as the wife of Thomas (m. 2 in 1597 to Higham, d 1599, French-2495)
Richardson claims Thomas Burrough (Burroughs-641), Gent., married (1st) to Elizabeth Burrell, daughter of Thomas Burrell of Dullingham, Cambridshire, by whom he had two children. Elizabeth Burrell doesn't currently have a profile.