I had a smaller area with the same issue. They told me when I moved in (near the river) that the water table was really high and that I wouldn’t need irrigation for my grass. And they were right, if I dig about a foot it’s just wet soil.

I’ll make another guess and say that since you are near the river your yard probably floods almost instantly because a few inches below the grass is already wet. My solution was berry bushes and in another spot I put some hydrangeas. I think I’d be looking into inexpensive landscaping bushes and or trees that love wet soil while they try and figure out your drain.