I kept having the same kind of problem in my old location. There was a lot of sand around I kept getting sand in all my drains. Anywhere I put corrigated or french drain, the bottom third of the pipe would fill up with sand and flow rates would just plummet. My situation was different though. I replaced most of my corrugated with straight clean PVC and give it enough drop the the sediment went with the water. Of course I was going to an NDS drywell so the angles will whatever I wanted them to be.

If you can figure out a way to stop the erosion, digging it out thoroughly , gravel, sock pipe, gravel would be great. In the same vein if you could get erosion under control dig it out gravel French drain gravel would be quite reasonable.

But if you can't get the sediment under control any type of underground conveyance is just going to eventually get plugged.

I would suspect the proper solution would be to regrade what's there with a retaining wall, but that's edging more out of DIY and into getting a proper professional to tell you what angles to actually use.

For a quicker cheaper fix, You might want to just construct something that's easier maintenance. Do a concrete gutter then you could just blow it out with a pressure washer.