Wow. It definitely looks and sounds like you have a water and soil problem. Here are a few thoughts and suggestions.
I think you were on the right track with your negotiations for a french drain. You may be in an area with a high water table in the winter. However, it was approved for building, so it should not be a problem this late in the spring when everything is growing. Having more plants and trees would help with transpiration, but they can't grow in standing water unless they are wetland species. I suggest you dig a hole up to two or three feet to see if you are hitting the water table. You should not be at this point in the year.
Here is a link to Clackamas County - https://extension.oregonstate.edu/location/clackamas-county-extension
. The pictures look like the grass is being water logged due to the lack of drainage. That is my guess since it is new sod. However, there could be other problems causing the discoloration and die-off that a specialist could identify. I would seek another opinion.
Contractors have a tendency during home construction to use heavy equipment that alters/removes the topsoil and/or compact soils, unless the yard was roped off to keep heavy equipment out of it. I am not sure if this happened, but it certainly is a possibility. If the ground is compacted, then you will see puddles on the surface and it will drain slowly. Clay is easier to compact than sand and silt. Given your pictures, it appears that you have compacted clay soils or so much clay at the surface layer that it is draining really slow.
The timing on the sprinkler system appears to be set to deliver small amounts of water throughout the week. A general rule of thumb is 1 inch/week total for the beginning and end of summer, and 1.5 inches/week during the hottest time. This is to compensate for more transpiration. I suggest you place a bunch of small cans, like tuna cans, around the yard to measure how much water is being applied. This is the only way you will know how much you are applying per day/week. We generally do not receive enough rain during the summer to eliminate the need for watering turf grasses without supplemental irrigation. It appears that the problem is with the drainage and not over-watering.
Given the slow drainage in the hole and color of the soil (gray), it looks like you have a lot of clay at depth. How long did it take to drain from the hole? The depth of the hole (e.g., 12 inches) divided by the time in hours is the infiltration rate (12/24 = 1/2 inch/hour). Filling it three times and timing the drainage will give you an average infiltration rate. This is a handy description when communicating with someone to help you fix the problem, such as a contractor to install a french drain.
Do you have any slope to your property that provides a natural drainage? If you build a french drain then you will need an outlet that is approved by the city. Usually people connect the french drain to the storm drain at the road or to a low spot in their yard that will absorb the water. You can't just send it to your neighbor, and it appears that your back yard does not have a good drainage area. You might be stuck with sending it out to the road.
You could haul in top soil and compost to improve the upper layer of soil in your yard for grass and shallow rooted plants. However, you will still have clay beneath those layers. If your property ends up draining and your water table is 3 feet or lower, than you can plant species that do well in heavy clay soils. Nurseries, Master Gardeners and landscapers can help you identify species to plant in these conditions.
This is a fairly complicated situation. Feel free to contact me by phone (503-510-7582) or email to discuss this in more detail. In summary, it appears that you have a lot of clay, possibly compacted top soil, and a drainage problem that may require a french drain. Please seek another opinion before you start investing in a french drain, because it will get expensive. I suggest you chat with the Master Gardeners and possibly Alec about the lawn and drainage first. Then seek out a reputable landscaping consultant for an opinion. There are a few large ones in the Salem area that have experience with these issues.
Good luck with your projects.