You can do a puzzle on any flat surface, but there's always the problem of how to glue the finished puzzle to that flat surface without turning the whole thing

You can do a puzzle on any flat surface, but there's always the problem of how to glue the finished puzzle to that flat surface without turning the whole thing upside down. After all, it's the backside of the puzzle that you want the glue to be on. I found that a coating of quick-drying, low-viscosity liquid glue on the front side works pretty well, because the glue easily flows into the gaps between the pieces, and then into the space between the puzzle and the board. Once the glue dries off, all the pieces and the backing board become a single chunk of cellulose-infused plastic. This is probably much better than either the thick mod podge or the adhesive sheet that OP tried to use at first.