"We know William had a son Thomas born in 1641, but is there any proof the older Thomas [b. 1617] (often said to be a brother of William) actually had a son Thomas at all?"
Yes there is pretty strong proof.
Thomas b.1617 had three sons by wife Elizabeth Andrews.
..... Thomas b. 1643 (has one 67 marker Y-DNA test of his line)
..... William b. 1645 (has five Y-DNA tests [1@ 67] of his line)
..... John b. 1647 (has no Y-DNA tests at this point)
..... All of these descendant testees have widely divergent and well developed paper trails.
The Thomas b.1643 descendant matches the descendant of William b. 1645 at 66/67 markers. So we are pretty sure we have the Y-DNA profile of immigrant ancestor Thomas Harvey b.1617 identified.
Without getting into the DNA weeds too deeply I think this is good support for there being a Thomas Harvey b. 1641 and a Thomas Harvey b 1643, both born in Taunton only two years apart. One line descended from William Harvey b. 1614 and the other from Thomas Harvey b. 1617.