It is the end of the day and the brain is not working at peak, but here are a few thoughts.
Do you know who else was on board the ship and where they came from? Did they all have the same set of political or religious beliefs, e.g., Puritans, Royalists, etc? Could either aspect narrow down John's origin in the British Isles?
Was John Bay traveling with anyone? I guess 16 wouldn't have seemed as young back than as now, but it seems young to be traveling by himself. If not, was he part of the crew?
Would someone from Glasgow or Dublin have to go to Gravesend to leave for the American colonies back then, or were there closer ports? Since he left from Gravesend, are the probabilities higher that your ancestor was from London?
Dumb but basic question: were the "way back" matches on men also named Bay/Bayes, etc? Our surname matches a few that are different, even though it was a y-DNA test.
And you might be into the realm of history rather than genealogy. I mean you might need to look for references in literature, historical writings, etc., for men named Bay in London, Dublin, and Glasgow. Court records? Parliamentary records? I don't know what is available.
Leaving at age 16, you probably wouldn't find John himself, but maybe a trace of the family. Any indication of their economic or social state?
Has anyone done a one name study of Bay?