Yes indeed, this family narrative has some inconsistencies. I have found an online copy of:
Notes on the history of the church and parish of Rattlesden, in the county of Suffolk : together with a copy of the parish registers from 1558 to 1758, and index of the marriages.
Olorenshaw, J. R. (1900)
On FamilySearch. [https://www.familysearch.org/library/books/records/item/102517-redirection Image link]
The transcript of the parish registers start on page 220 (image 253). The registers start in the year 1558.
Henry does not appear in the register for 1560.
The first mention of the name Scott in any variant is in 1567 (image 258):
Dinay the dawghter of Edmund Scote and Joane his wife baptised 18 June 1567