Caleb - in reviewing your soil test results I see that the soil pH is fairly acidic at 5.8.  Most lawns do best in a pH range of 6-7.0 at which there is an increase in the availability of nutrients.  I am including links to numerous articles about pH and applications of lime.  The first one you may find most helpful in understanding the need for a higher pH and how to apply the lime product. As you will see, applying lime can be done throughout the year rather than all at one time.  It will take several years for the pH to change.  Other than the low pH, your soil looks healthy!

https://ohioline.osu.edu/factsheet/hyg-4026

https://www.canr.msu.edu/resources/facts_about_soil_acidity_and_lime_e1566

https://www.canr.msu.edu/news/making_the_most_out_of_your_soil_test_ph_factsheet_now_available

https://www.canr.msu.edu/uploads/files/Soil_pH_Finnerann.pdf

Good job starting with a soil test to know the health of your lawn! Hope that helps!

My lawn is heavily shaded by several large oaks and maples around the perimeter. I have core aerated and overseeded at the optimal time the past 3 autumns (with starter fert and plenty of water). I’ve used a shady mix twice, and most recently Jonathan Green Shady Nooks (the most shade tolerant seed I could find). I have never seen a noticeable difference from the overseeding and my lawn is very thin. Every late-spring it thickens up and looks promising, and then gradually thins out the rest of the summer/fall. I water twice per week for a total of about 2/3 of an inch per week, and I fertilize at approximately the nitrogen rate recommended in my soil test results. 

What am I doing wrong, or what can I do differently to get a thicker lawn? Is it likely the low Ph is causing this, or is it most likely just the shade? One particular thing I’ve noticed starting in early summer every year is that there is healthy grass and dead grass mixed together - maybe that indicates a specific issue. Pictures attached. Thank you. 

My lawn is heavily shaded by several large oaks and maples around the perimeter. I have core aerated and overseeded at the optimal time the past 3 autumns (with starter fert and plenty of water). I’ve used a shady mix twice, and most recently Jonathan Green Shady Nooks (the most shade tolerant seed I could find). I have never seen a noticeable difference from the overseeding and my lawn is very thin. Every late-spring it thickens up and looks promising, and then gradually thins out the rest of the summer/fall. I water twice per week for a total of about 2/3 of an inch per week, and I fertilize at approximately the nitrogen rate recommended in my soil test results. 

What am I doing wrong, or what can I do differently to get a thicker lawn? Is it likely the low Ph is causing this, or is it most likely just the shade? One particular thing I’ve noticed starting in early summer every year is that there is healthy grass and dead grass mixed together - maybe that indicates a specific issue. Pictures attached. Thank you. 

pictures attached

Caleb - shade is not your friend with turf.  With heavy shade, the pictures you have included are typical as the summer progresses.  I am including a link below with more info about how at least 8 hours of sunlight are needed for a healthy, vigorous lawn.  You may want to consider over seeding with fine fescue which is more shade tolerant that other grasses.

https://www.canr.msu.edu/news/will_grass_grow_in_my_shady_lawn

Hope that helps.