There is a time and a place for a french drain. I am of the opinion that this isn't the best fix for this area. There are others who disagree with me, that is fine. Here is my thought looking at the pictures and looking at the problem, it isn't a slope issue, it is fence and a mulch issue. Beyond the fence there is evidently french drain system in place. So the drainage was apparently don't correctly before you were called to the site. The problem happens with the mulch is carried to the fence and plugs up the outlet. The idea of clearing the path of the mulch and adding fabric under river rock to create a dry stream effect. There is already rock back there, so the dry stream bed will not look out of place and will fit in really nicely. The advantages, you are using the surface to your advantage, it is already there. Next you dig up a yard and drop a franch drain which is well and good, but you have to dump the water out somewhere. I am guessing that will lead an outlet lower than drains on the other side of the fence. That is creating more work and headache for you.