I'm not sure how to get all this into Research Notes. But here are all the relevant entries, along with the image of the Sherborne PR where they can be found:
Oct. 30, 1591, Thomas Gardiner, bapti, Image 388
June 5, 1598, Margaret Ffriar, bapti, Image 398
April 28, 1617, Thomas Gardiner et Margaret Ffrier nupti, Image 424
March 8, 1617[/18], Thomas Gardiner filius Thomas Gardiner bapti, Image 425
Jan. 1, 1619[/20], George Gardiner filius Thomas Gardiner bapti, Image 427
July 20, 1622, Richardus Gardiner filius Thomas Gardiner _______(?) bapti Image 431
Dec. 7, 1624, John Gardiner filius Thomas Gardiner menori(?) bapti, Image 435
July 18, 1627, Samuel Gardener filius Thomas Gardener bap, Image 439
Feb. 6, 1630[/31], Sara Gardiner filia Thomas Gardiner bap, Image 444
Mar. 24, 1632[/33], Miriam Gardiner filia Thomas bpt, Image 447
I have found no record for Joseph, probably because many of the lines are illegible, and the record for Richard, I think must refer to a child who died young, as Thomas' known son Richard died in Nantucket in 1724 at the age of 92, indicating his birth to be around 1632. That baptism could also be in the records but illegible. Overall, though, I think this is pretty convincing that the Thomas Gardner who attempted the settlement at Cape Ann in 1624 and then became one of the first settlers of Salem was different from this one. I have documented every Gardner/Gardiner/Gardener and Ffrier/Ffriar I could find in the Sherborne PR from 1572 to 1637; there are a total of 90 Gardiner and 10 Ffrier entries, bapti, nupti and sepul, so from these records I think it is impossible to infer any parentage of Thomas, or even Margaret. Before 1604 no parents were named. There are several Thomas Gardiners who were baptized, married and buried throughout those years. But the only other family who seemed to be having children at the time was that of a Robert Gardiner. I don't have much doubt that all the baptisms naming Thomas as the father were for our Thomas.