I have some updates on this question.
1.
Two newspaper columns from 1821 detail a meeting of the Suffolk Pitt Club.
Thomas Sherlock Gooch is the president and Henry Gooch is a vice-president.
There is nothing to say which Henry Gooch he is or what family connection he might have, if any.
The Henry Gooch of my family would have been 35 at the time.
2.
I was surprised (embarrassed?) to find that I posted this very same question on Rootschat nearly two years ago and then very quickly forgot that I had done so. Within a day of putting up the question in Jan 2017 I discovered a Gooch family tree in GENI that listed dozens of relatives to the Baronet family but included none by the name Henry Gooch. At the time I took this as conclusive evidence that there was no connection between my Henry Gooch and the Baronet family.
3.
However, I have lately come across a book written in 1802 or thereabouts which gives a very thorough account of the history of the Gooch Baronets up to that time and finishes with a hand-drawn family tree that includes some relatives not listed on the GENI tree. Two of these are Henry Gooches, descendants of people mentioned on the GENI tree but whose families were not explored. The two Henrys would have been alive at the time of the 1821 club meeting and aged roughly in line with the Henry Gooch of my family tree. But they are not perfectly aged for a match. In both cases the men would have to be about ten years old at the time of birth of my own Henry Burton Gooch.
4.
Having said all that, it shows that the GENI tree is not complete and it is possible there may be other places where a connection may be formed.
To be clear, I'm interested in seeing whether a connection exists only insofar as it would explain how my own Gooch ancestors became so wealthy and were able to send their sons off to
all the corners of the (then) British Empire. It would also explain how 'Sherlock' became a middle name passed down a couple of times in my Gooch family. And it would extend the horizon of my Gooch family tree by several generations, which is always satisfying.
Any assistance welcome.
Cheers
-David C