Sometimes, it is better to try hardscape in areas where grass will not grow.

Steps from the deck, gate through the fence and kids toys all speak of traffic. Is there a dog too? Plants and traffic are at odds with one another.

Make a defining edge between where the grass will grow and where it won't. Cover the dirt in pine straw.

I agree you should edge the grass, and turn the rest into either an informal mulched area with stepping stones, or a gravel patio.

The gravel patio can include a sitting area with a fire pit, shade-loving shrubs in containers, etc. Much better than grass! But I'm not sure if that would work for the kids and their toys...

Another vote for don’t fight nature. Edge the lawn neatly so the grass-free area looks intentional rather than bare. Then deal with the sandy area with some kind of mulch. Please don’t even think of sacrificing the trees for grass.

Have you tried a groundcover for dry shade? Geranium macrorrhizum might work, a handsome, no work, deer resistant groundcover..

Given the obvious existence of active children I’m not sure how any plant will work there. I agree with Chris. You can’t pedal a toy car through a ground cover.