The drain needs to empty into a dry well / seepage pit. (Google the terms, there are many images.) As to the clogging, the only answer is to remove the drain cover before you leave town so the solids (leaves) will flow along with the water. Increasing the size of the drain AND the "slope" of the pipe leading into the dry well might also be needed - dependent on the amount of rainfall you receive.

I would NOT remove the drain cover! There's a very good reason to keep junk out of the drain. And something big and flat that blows in can still block the drain.

What I'd do is get one like they use on roofs--one that stands up, like an inverted strainer. That will keep your drain clear and keep pipes free-flowing.

I'd also take out the bricks that block access to the stepping stones. Even without the worry about ponding, they make no sense there. Also, if any downspouts lead to the patio, divert them. And put a diverter/roll-out extender on the downspout that is visible in the photos.