I have a paper trail back to the the family of Col Thomas Waite, the Regicide, and the lines preceding him back in the UK. The links include names Thomas, John, William, Richard. This may be some use if trying to peg your line back to the UK.
Tree wise, my direct Waites go - Dorothy Waite b. abt 1681 m. John Balch, her parents Roger Waite b.1659 Market Overton, Rutland, England and Ann Williams (links to Jamaican family of Raines Waite, Barretts etc.),
Col Thomas Wayte born 1615-Wymondham Leicestershire d. Jersey 1688 and Jane Raines (dau of Robert Raines of Stanford on Soar, Nottinghamshire),
Henry Waite b. 1593 Wymondham Leicestershire and Susanna Gulston (Wymondham, Leicestershire),
Thomas Wayte b. 1563 Wymondham and Elizabeth,
William Waight or Waite abt 1538 d 1568 Wymondham, Leicestershire and Margery Redford,
Richard Waite-his father born about 1510. This is interesting as I see Richards in your mentions.
Went through lots of National Archive Papers in the UK and Wills to confirm through sibling lines, Leicestershire and Rutland records. Family links mentioned to Cornwall, Wiltshire, Suffolk, Jamaica, Warwickshire and other counties in the UK.
If you get to a point where you are interested in trying to link the lines in the US to these lines, let me know.