I am a big fan of drainage around the house but it's a challenge as water will go where it wants to go.
Yes, if you backfill with gravel and top with more landscape fabric to keep out the soil then turf it will drain better for the area on top of the French drain.
I recently learned that in heavy clay areas contractors prefer to use pvc pipe with holes drilled in as flexible drain pipe will collapse if ground pressure is applied such as from vehicles or pedestrians.
I suggest using a level to see where the slope is. Laser levels are readily available and if you use them at twilight the little red dot is much easier to see.
If you are going to go to the trouble of digging trenches, laying fabric, 4" PVC flexible drain pipe with slits you might as well go all out and check the levels near your house. If required top dress or change the grade so water will shed from the eaves or drain pipe and go towards your French drain.
And, if you think there is even a remote possibility of water entering your house through basement windows put in window wells with a dry stone well beneath the well and grade away from the windows. An old plastic bucket with the bottom punched out works great.