Back yard get's very soggy after rainfall

I have a sloping back yard that levels out into a wooded area. The yard is about 100ft by 60ft The last 10-15 feet half of that yard (the lowest lying part) gets very soggy after rainfall and stays that way for a while. My theory is that all the water travels downhill (both surface and subsurface) to the back of the yard and then takes a while to absorb because the soil is not very porous.

I was thinking of putting in a 3-4 ft deep french drain filled with gravel (pipe then gravel all the way up with the last 6 inches being soil for grass) a few feet uphill of the portion of the yard that gets soggy. I'm thinking that this will catch most of the surface and subsurface water that is making its way downhill and this will get rid of my soggy backyard problem. I would daylight the drain into a culvert on the side of the property and also potential a deep dry well (thinking of using the NDS Flo-well).

Would love to hear the communities thoughts on this idea.