Designing buildings is somewhat like software development without type checking, unit tests, and compiler warnings - and that's for professionals with good tools and a formal design process.

Looking at the pictures my advice would be to keep the roof free of debris as the first strategy. Second strategy is to make each drainage path independent, I.e. don't dump water from one roof/gutter onto/into another roof/gutter. Third strategy is use primary and secondary drains. secondary drains should do two things, handle excess load and make their use obvious to signal problems with the primary.

The screens on the lower gutter may make your problem worse due to a lack of scouring and creating a dam at the top of the gutter rather than the bottom. Better to reduce the volume this gutter handles.

Finally the goal is always water management not water proofing. Assume the first system leaks and handle those leaks with a backup. And of course never depend on water running uphill.

Good luck.

A "fix" I see with some regularity (especially for flat roofs on mobile homes) is to build an entirely new roof above the leaking roof, supported by long poles driven into the ground surrounding the structure. This may not be feasible for you, but it certainly seems to be for people around here. The additional roof is very simple sheet metal, and isn't even heavy enough to require trusses. In many cases it doesn't even require a peak, being sloped all in the same direction.