Hey, I am very familiar with this soil. Either way, the problem is that when the construction crew built your sub community they dozed the top soil to clear the
land. Your soil is very dead. It will take some years for the grass to grow thick - around 3 to 15 years naturally. Due to your slope every time it rains you will have soil erosion removing the nutrients from the surface further prolonging the fertility regeneration, hence the bald spots of grass clay. Also the clay has a tendency of compacting.
You can plant trees but they will grow very slowly because the soil is not very fertile. I do not believe you are allowed to use many
permaculture techniques involving animals or anything of that sort in your subdivision so conventional landscaping would have to be considered. You could directly transplant patches of grass from your
lawn but this is very time consuming and you have to work in some
compost or fertile soil into the clay before you place the grass. Make sure not to do this before heavy rains due to the slope. The grass will be in shock but will bounce back and since the soil is clay you you dont have to worry so much about
water.
Be careful not to be tempted to do the seed thing, when the fall and spring come the hardiness differences in the grass (if lucky
enough to germinate) will make lots of brown spots and it will look very ugly as well - not much improvement. Also if the back yard is frequented a lot you
should consider a shrub or tree more because people make more forceful steps walking at angles which will kill the grass.
BUT IF I WERE YOU
I would not focus on monoculture grass lawns and focus on building functional herb guilds, maybe a mixture of
medicinal plants like and cluster them together. They will spread and I am sure you will appreciate going to the back yard for clippings of fresh herbs when you cook.