We built our rectangle pool in grass a few years age on CT. The builder we hired usually works for well known local pool companies. Most aspects of the project were clearly in his wheelhouse but others were not and that probably had minor impact on our final product. That said, we paid half as much by working with him.
Our town has strict drainage regulations which the pool builder was not involved with. I’m not sure if this is a factor in your area but if it is, be sure you know those costs too (engineer reports, cisterns, rain garden etc.)
Our coping is bluestone. It’s a lighter shade and I don’t know the name. Strangely, the corners are a darker shade. Maybe it was easier to shape the corners. I don’t find the coping too hot. It’s actually cooler than other decking options in my opinion. And, the grass is AMAZING! It always feels cool, clean and soft on the feet.
Grass in the pool hasn’t been too difficult to manage. For the previous two summers, the lawnmower had a bag so grass clippings were removed. The machine had to be run so clipping blew away from (not toward) the coping into the bag. Now the mower minces clippings and leaves them to mulch. This has not caused more debris in pool. After mowing, we use a weed-eater around the edges. This gets a bit of grass in the pool but the skimmers easily handle it. Use skimmer socks to minimize the amount that flows to the filter if you like.
If you fence the pool, be sure the gate is wide enough to accommodate any lawnmower that will be used.
Something to keep in mind is where/how your groundwater will flow around the pool shell. Due to slope in our property, we had water flowing underground by the deep end. In addition, I kept the solar reel there so extra water was being introduced from above. It was a bit soggy and we developed a dip beside the coping which I feared would get worse. It was also an ankle twister. This year we added a brushed concrete patio by the deep end with built in drain. Now I wash off solar cover here, remove the robot etc. No more water issues.
Another thing to keep in mind with grass is location of patio furniture. Prior to the new slab, I kept all furniture in a shed and pulled out chairs as needed to avoid sprinkler issues (spraying furniture/cushions, blocking irrigation).
We opted to skip the auto cover due to both initial cost and potential cost of repairs. Rather, I have a solar blanket on a reel. I could replace those 20 times and it would still be less $ than the auto initial cost.
Around here people seem to use the mesh cover in the winter even if they have a solid safety cover. Our safety cover is attached into the grass. This has not been problematic in any way. Cover stays secure.
I hope this is helpful. Again, we love the grass!!