Yes, I have used a paddle board and quite successfully. But not all paddle boards are the same. The one I used was rotomolded and had bungee webbing on the tail/stern. It also had webbing up front on the bow area. I placed my aluminum 63 rigged on the stern and secured it with the bungee, partially inflated wing. My lifejacket up front on the bow and a mesh bag (lobster type) with my mask and fins. The one I used was similar in design to this, it may have even been this one:
No Limits Aries 12' Stand-Up Paddle Board | Academy
I paddles with a kayak type paddle from a kneeling position. That could be

a balance issue for me since I am not but a beginner paddle boarder! Concerning the bungee, I would rather secure the SCUBA with two 1.5 inch straps with QR buckles. The bungee is not secure and is a PITA and the gear will spill from it should you capsize (and you probably will). I used a Pelican reel to stow my tow line and simply towed it behind me as I often do with my kayak, I did not anchor it.
Now, if I had to make a choice between a sot kayak (properly rigged) and a paddle board I am taking the yak but I see no reason that, depending upon the paddle board, construction and size, that you might be able to dive from it. This is the thing, way, way back, I had a dive board, surf ski. This was long before rotomolded kayaks or paddle boards. I used it for free diving and while it did not really have the capacity, more than once I SCUBA dived from it. I put my tank on, laid on my belly as was the style and paddled with my hands.
Good luck, hope it works for you.
N