You aren't going to like my answer but grass will not grow under mature maples.  It is too shady.  You can try some fescue but it will not be the 'lush' lawn you are looking for.  You could also try planting shade tolerant ground cover.  Personally,I have two large maples and gave up on grass many years ago.  Also, the roots will come to the surface in spots and it is not a good practice to cover them up because they will just come to the surface again.  I have mulched under those trees and have some areas with attractive hostas, in various sizes and colors.  Also, some fun decorative containers and a birdbath.  Think of it as less grass to mow.  https://extension.umn.edu/planting-and-growing-guides/gardening-shade

With the additional information you have given me, especially about the amount of sunlight available, I have taken a second look at the situation.  What you  are probably experiencing is called Dry Shade. It is a situation where the tree takes up the majority of water available because of the extensive root system.  You could try additional deep watering in that area.  No need for an entire irrigation system. A hose end sprinkler could be used to cover the area needed.  It probably isn't a matter of nutrients but, if you wish, you could apply some slow release fertilizer in the area too.  Yes, it would be a good idea to do a soil test.  https://soiltest.cfans.umn.edu/  You could try these applications to see if the grass improves.  Do not bring in more soil.  That will harm your maples and could lead to stem girdling roots which will kill them.  Make a point of actually keeping track of the sunlight that hits the ground in various areas.  It may be there isn't as much as you think.  Good luck.  As I mentioned to you before, I have just mulched the area beneath my two very large maples and am glad to have removed the frustration.

There are grass seed mixtures available at local garden centers that are labeled for shade/partial shade which will include a mix of fescues.  Take a look at our instructions on dormant seeding.  That would be the easiest way to introduce new varieties.  No need to dig up good grass. https://extension.umn.edu/lawn-care/dormant-seeding 

Good luck.