I’m currently using 1/2" MDF. Seems to work pretty well, all the puzzles get primed and painted and cleared. The pockets are machined using a 1/4" bit and then a detail pass with an 1/8" so it doesn’t take forever.
I’ve started look into the method you described using a 1/4" piece of plywood and gluing it to a 1/2" piece of MDF, but it seems to take the same amount of time have just the MDF. Where I need to figure out how to save time is finishing the letters.
I think I posted a while ago about my puzzle process, I’ll try to hunt it down and update this post. I hope this helps! I’d love to here if you come up with something that works well, I’m always looking for tips and improvements!
The only thing I’ve changed from the process stated below is I don’t use steel wool much before the very end. The steel wool sheds fibres when you use it and I was getting pieces in my finish.