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Start from what the cleat is actually doing, because it is not carrying the load the way people assume.

The pair of angled strips converts the hanging load into two things: a downward force taken by the bearing surface, and a component pulling the wall-side cleat away from the wall. The critical failure modes, in order of how often they occur:

1. The fixings pull out of the wall. This is by far the most common failure and it has nothing to do with your timber choice.
2. The cleat splits along the grain at the fixing line.
3. The bevel crushes or deflects under load.

So timber selection is mostly about resisting the second one. What matters is straight grain with no runout, no knots anywhere near the fixings, and adequate thickness so the screw has material either side of it.

The two halves of your question have different answers, which is why it is worth separating them.

For the screw: a screw holds far better going into the face or edge, across the grain, than into the end grain. In end grain the threads are running between fibres rather than biting across them, and pull-out strength is dramatically lower — commonly cited as roughly half, and worse in some species. This is the strongest single principle in the whole question and it applies everywhere, not just here.

For the load: wood is much stronger along the grain than across it. A support carrying a bending load wants its long fibres running along the direction of the span, so the load is carried in tension and compression along the fibres rather than trying to separate them.

Happily these usually point the same way: grain running along the length of the support, screws entering through the face.

What I would use instead, at similar cost:

Nothing. Treated pine outdoors, left bare, weathers to grey and lasts for many years. For a bin enclosure that is a completely reasonable choice and it is free.

A penetrating exterior oil made for the purpose. Costs more than free and less than most people expect, contains ultraviolet inhibitors and often a fungicide, cures properly, and does not transfer onto clothes.

Exterior stain. Best appearance retention, needs recoating on a schedule.

Given that you are not concerned with appearance, the honest comparison is between doing nothing and a proper exterior oil. Used engine oil sits below both once you account for the disposal issue, and it is only cheaper because the cost is being paid by the ground it drains into.

The lowest-technology answer, and on a small door it is genuinely competitive: a hand plane.

Plywood edges plane easily if the blade is sharp and you work from the outside in to avoid tearing the outer veneer. Any plane will do — a block plane is fine for this size of job.

The approach that works: rout or saw close, leaving a millimetre or so, and then plane the last bit down to the frame. Working by hand for the final pass means you can follow the bevel by feel, checking with a straightedge across the two faces, and there is no setup at all.

On a job this size I would be finished before I had the router table configured.

The corner is the whole problem, and it is worth being explicit about why.

An L-shaped desk is two panels meeting at right angles. Each panel is stiff in bending along its length and neither resists twisting at the joint on its own. When you lean on the far end of one arm, the load path runs through the corner, and if the corner is not supported and not stiffly joined, the assembly racks — the two tops move relative to each other and the whole thing feels wobbly regardless of how solid the panels are.

So: put a leg at or very near the corner. That is the single most important placement decision, and skipping it for legroom is the most common mistake in these builds.

After that, a leg near each free end, giving you three or four support points. The unsupported spans between them then determine whether the tops sag.