I have no intent to throw rocks at the MJ Sailing couple in any way. I do like their SBH-related videos. For me, it's always a pleasure to see the island through "fresh eyes".
We all have different skill sets and different knowledge levels. I wouldn't trust me to sail a toy boat from one end of a swimming pool to another. The MJ Sailing crew have much better sailing skills than I do, but they have never been to St Barth before..
I have some experience on many Caribbean islands, but my knowledge on them is limited and dated. St Barth is where I've chosen to visit for many years, and I have much more extensive knowledge - not encyclopedic, but better than most visitors.
Someone from this forum could have helped with almost ALL of their issues. Some could help with one or two issues, others could help with more issues.
One restriction that many people sailing on a budget have is that they have no wheels. Anything rental is a big budget buster, and being limited to dinghy or walking poses severe restrictions.
The MJ Sailing crew appear to have been on-island on Easter weekend (witness the beach tents, the only time when they're allowed). Even on holiday weekends there's always a grocery store open, for at least part of the day. But it's going to be a drive, unless the tiny Chez Alex/Epicereie de Corossol store in Corossol is open. (How many people here know how to find it, and how many would know that it is often open mornings and afternoons on a holiday, but with a long lunch break)?)
The hardware store? Tough to approach by water, but the walk to it from the port, dangerous as it is, is no more dangerous than the way that they tried to walk to it from Corossol, and did walk to it from Public. (Might have been able to find that gas bottle at Chez Alex too. I know that she carries outboard motor oil, so maybe propane bottles.)
Fuel system parts? Try the Ship Chandler above Le Repaire (might have been able to get the propane bottle that they were looking for too).
Anchoring at Shell Beach, I would have recommended the other end of the beach, over where Flag was moored - less chance of hooking the fiber optic cable with your anchor.
As to filling their propane tank, I've never done it, but I know that there are two places where it can be done - but not in walking distance from the port.
Much of the above should be in a good Cruising Guide which covers St Barth. I've never read one, so I don't know if it is there or not.
Again, no rocks intended, and I do like their SBH-related videos.