It is really cool to see that you plan to use such a well-conceived solution to the backyard drainage. I see many people suffer with chronic basement flooding because they do not recognize the need for a repair like yours.
Another helpful thing you can do is to grade the surface to slope a little, away from the house. Then install a 4-6 foot wide polyvinyl plastic sheet under that gravel located up against the house.
And install a 10-foot long downspout diverter to drain the downspout away from the house. (or hook the downspout up to your new drainage system).
All of the gravel has been removed and replaced with compacted fill and topsoil and overseeded with grass. I actually did install a temporary diverter to my downspouts to route water down to the backside of the house until the seed sets.
The slope of the rest of the yard (the yard under my feet when I took the photos) had TWO concrete patios buried underneath it. Someone years ago wanted a patio and poured one, then a subsequent owner didn't like it and buried it, then another owner wanted a patio and poured a new one, and yet another owner didn't like that one and buried it again (so it was soil/concrete/soil/concrete/soil, like a layer cake). I sorta wished concrete mix was more expensive so people thought before the built, but what are you going to do? Anyway, I had all of the concrete dug up and regraded, so the level of the yard dropped over 9 inches. It still is a bit high, but I can't lower it much more without it becoming lower than the neighbor's yard.
A french drain is still an option (and an easier one now that all of the concrete is gone and the grading makes its location easier to set). I want to see how things settle out after the grass sets. Things are already much improved now that I don't have a swimming pool right up against the foundation. The crawlspace was amazingly dry regardless of the previous bad grading, and I want to keep it that way.