Have just checked OH's Fairburn match (living in England) and it checks out to accuracy down five generations, the spouses and dates are correct.
Also checked if the match has any common matches to OH with the surname Fairburn and there are about 5 - all descended from Thomas Fairburn who left England for Ontario, probably as a young man.
So am pretty confident in saying Thomas is related to the family in Bridlington, but his exact connection is unknown, perhaps an illegitimate child who was not baptized (I have one of those, second child of a couple, first one was baptized, but he wasn't, though his father acknowledged him - perhaps the vicar refused to baptize unless the parents married!).
It is fairly easy to narrow down OH's connections to English branches as he is a mix of Danish, Swedish, Irish, Scottish, English, and back in the American colonial period, likely German and Belgian. His English percentage is not very high.