My wife's idea is to use window screen mesh (fiberglass) to serve as leave guard, but maybe the holes of these fiberglass window screens are too small. Any "mesh" materials that we can use that may have bigger holes on the mesh that can let water go through but will catch the leaves? Thanks!

My thoughts are like your wife’s. Perhaps a fiberglass or coated hardware cloth would work and they can be found with 1/2” mesh, like here: 1/2” coated hardware cloth

Any screening is just going to add another cleaning step to the process and will end up blocking the water from getting into the drain like Klem told you. To prevent that, you will end up needing to clean the screening more often than what you have now.

There are a variety of different trench drain covers on the market. I recommend you get a section and try it before committing to replace the entire cover. Here's one I would try https://www.amazon.com/Channel-Drainage-Driveways-58X25X2-5CM-22-8x9-8x1IN/dp/B0DN2159DJ?th=1

I chose the window screen and frequent clean method on a drain at my previous home. It sure beat having clogged pipes. I just considered it a part of yard work.

Contact the manufacturer of your existing cover. They could have an exact fit alternative.