The southernmost launch in that Boatlaunch listing is the Burnham-on-sea "slipway". This is actually an old sloping rail/steamer pier that passes as a slip these days. Charge inc parking is £25-£30 a day for launch/retrieve and it really just gets you over the short sand element of the beach and onto the water a bit either side of high tide and gloopy mud otherwise.
This Google street view below gives an idea how far out it dries if you leave the tide a bit late, as far as the eye can see. My link should land you on the Jun 2011 view, if not change to that to see the lower tide situation.
The more northern launch location is unknown to me but I think it's a boat club or business slip into the river Axe which dries totally for much of the day.
Re the whole idea of course everything is possible but from anywhere in that area you will need to passage plan so you have a mud free launch/return and that there isn't too much water at Flat Holm when you arrive (some of the landings there cover at higher tides). Watch out and allow for the fierce tides that will set you up/down from the planned heading. Allow for the surprising speed of the large shipping when assisted by tidal flow. Unless flat calm watch wind over tide in the fierce currents flowing around Flat Holm. Have a good plan for engine problems as the tide will move you a long way over even a short period faffing about sorting an issue.
The folks that do seem to be about there in groups are the sea kayak guys who seem to love the challenge of getting out to The Holms.
https://maps.app.goo.gl/jTrmBfXqAdKL6hKS8
