We made French drains for our sink water and our washing machine water.
We did not use any plastic.
We did not use any pipe or landscape fabric.
We dug a trench along the foundation of our house that we filled with rock, then covered with dirt.
We have been using this system for almost ten years.
Similar to this:
https://permies.com/t/1356/french-drain-pipe-water
Erica said, "If you don't have a pipe, the gravel still allows water to flow out as long as your grades are right, but it can't move as much water as quickly. Your "Paul" drain would probably work; just not as well as the conventional drain.
https://permies.com/t/1356/french-drain-pipe-water#8350