I built a bookcase from 3/4" (19 mm) MDF back in 1985 to hold heavy, oversize college textbooks.
It's three feet wide with two twelve inch deep, fourteen inch tall shelves.
It has not bowed under the weight in 27 years.
The shelves are are attached to the side uprights with four screws on each end.
The back is 1/8" masonite, well secured with screws. 1.5 inch rails attach the end boards at the bottom also at the back of the top.
Judging from the stability of the bookcase under heavy load for a quarter century, I'd say that dados are not necessary.
Use waterproof glue, make the joint tight-fitting, and leave the screws on the shelf. A lap joint is one of the strongest of all the glued wood joints, and a well-fitted glued lap joint will be at least as strong without the screws as with them.
But... there's a condition. How wide are the two planks forming that joint? And which wood is being used? I ask because of wood's inherent expansion/contraction with humidity/moisture changes. If the joint is too wide for that species of wood, and the wood isn't extremely well sealed against moisture intrusion, then expansion/contraction will destroy the joint; nearly all woods expand & contract across the grain far more than they do along the grain. As the wood becomes moist, it'll become wider but not longer.
Certain woods have a very low expansion/contraction rate. Other woods have a very high expansion/contraction rate.
If the wood is extremely well sealed, its inherent expansion/contraction will be minimized, but can never be totally eliminated (there's no such thing as a perfect seal).
If you're using wide boards, and that species of wood is subject to high expansion/contraction, then neither glue nor screws will prevent the joint from failing. Better in such a case to make a different type of joint, like a dovetailed saddle joint pinned across the "inside" corner, for example. That's much more difficult to make, though. Another option would be a long mortise with a tenon narrower than the length of the mortise.
Good outdoor gate joints are difficult things to make, since they're exposed to the weather and tend to collect pockets of water after a rain.
Best Answer
Glue can be pretty strong when you've got plenty of area (like there), although unless you've got some diagonal bracing, it's going to be wobbly whatever fixing you use.
Most people don't use glue because it takes a while to dry, so you either run out of clamps after a couple of joints, or the clamps get in the way of the next thing. (by comparison you can bang in a few nails and call it done)
If you're worried about the screws rusting, use screws as clamps, and remove them after the glue has dried. Or leave them in and let them rust.