I can say with conviction that the William Owen buried in Williamsburg was not likely related to Bart or the William who came to Virginia in 1650. Pretty sure he was just a guy. The record for this guy in Williamsburg suggests he was an inmate at the asylum in Williamsburg - which means he was either batty or a total alcoholic....I've visited the restored asylum; would hate to think an ancestor lived in it!
And, yes, I agree that " Do we agree that Sir William Owen of Shrewsbury is probably a different person than the William Owen of Machynlleth?" The thing that gums the whole effort to find out which Owens belong to whom is that I think Sir William Owen of Shrewsbury is descended from a Richard Owen who did go from Machynlleth to Shrewsbury in the 1500s and may be related to some Owens who stayed in Machynlleth. I had thought at one point that the William who came over here was the son of Sir William - but he wasn't. I found that son quite dead in the UK. Not our guy. My hypothesis is now that William who came here was from an indirect branch of that family.
The best way to settle some of this - and get most of the craziness off of all the online genealogies - is to find men still in the UK named Owen who carry the same Y-DNA as the guys who came over here. Go visit http://www.hprg.com/owen/ You will see that the descendants of Bart carry the Y-DNA of R1b-L2-FGC46807. This means that Bart's father and grandfather and all the way back had Y-DNA of R1b-L2-FGC46807. Now, we could go over to the UK and dig somebody up and see if we could get DNA out of their very old bones, but....
The descendants of the William who came over here in 1650 carry the unusual Y-DNA of J2a1a-BY32938 so we know that 1650 William was not the father of Bart, who appears to have come to Virginia in 1619 - nor Bart the father of William.Who were the fathers of William and Bart? I think we don't know!! All we know is that they weren't related to each other - and that both were likely Royalists/Cavaliers, not friends of Oliver Cromwell! I gather people still in the UK don't do DNA tests like we do because they know who their ancestors were! And they don't seem to spend much time worrying about who got on a boat and went to the colonies.
I had in mind to just go over to the UK and start digging in the archives at Shrewsbury and Machynlleth - and then along came COVID. And then along came inflation and wiped out my retirement money. So, if you're still financially sound, get yourself over there and start digging. My 5th cousins from Georgia(we both descend from William, not Bart) went to Machynlleth and Shrewsbury pre-COVID but didn't find any smoking guns. My brother also went to Machynlleth a few years back but said the only Owens he found were in cemeteries - just the ones we need to get DNA from!! He also said the brew in the pubs were excellent.