I'd run a 2 X 4 or 2 X2 along the top of the long rails, flat to the top. Screw and glue--you'll have long-grain to long-grain which is ideal for glue--to the rails with a good, yellow wood glue. Then you can either use screws short enough they won't come through the top from below--that would be my preference--or countersink and screw from above. If you want to be able to easily remove the top in the future, don't glue it to the rails. I wouldn't fasten it at all on the short edges, and keep my fasteners back about 1/4 the length of the top from the ends to allow for expansion and contraction of the wood.
Iron-on edging will work much better than nothing and it's easy. If you have a router, you can trim it flush with both sides, if not, put any excess on the bottom of the edge.