You have tried grass and it has failed a couple times. That sounds like you do not have enough sun to grow it; typically you need at least 6 hours of sun (grass struggles along), with 8 hours it will do better. Other factors that encourage moss are compacted soil, low nutrient levels, and moisture.
If you haven't had a soil test, you should take one before choosing plants or trying grass again. Some shade plants will have a preference for a certain pH and soil type. You can purchase a soil test self mailer at MSU extension office, or online (www.msusoiltest.com
If you need an area for children to play, or people walk on regularly, then ground covers tend to be a poor choice. You could have trees and shrubs properly pruned or trimmed in the area that gets the most sun, and seed a shade tolerant grass like tall fescue in just that area. Correct nutrient and compaction problems before reseeding.
Moss gardening is becoming popular. Mixing other shade loving plants in garden beds and featuring the mosses is one idea. Enter "moss gardens in Michigan" in your internet search to find resources and pictures of these gardens.
There are a number of ground covers that grow in shade and part shade. Enter "lawn alternatives for the midwest" in your search engine to see ideas. I have included some links, below.
Mulched areas around the trees can help cover some of the problem area. Mulch is better for the trees than lawns or ground covers because mulch doesn't compete with trees for water or nutrients. Keep mulch pulled back from trunks by 5-8 inches. Pathways from house to garage, etc. can be mulched or paved.
The remainder of the area can be planted with shade type plantings, from ground covers to shrubs, perennials, even small shade loving trees.
Here are some ideas/resources-
Sedge- some can be grown in Michigan- Carex montana, Carex glauca, Carex conica, Carex digitata are some low growing sedges that are hardy to USDA hardiness zone 5. They can be mowed to about a 4 inch height, if desired.
Festuca grasses, such as turf-type fescues, are more tolerant of shade than Kentucky Blue Grass.
You are located in plant Hardiness zone 5-6. Select plants that are at least zone 6, zone 5 or lower would be best. Here are some resources for shade plantings:
Please write us again as you develop a plan, and have more questions.