Mark - in reviewing our soil test results, I see your soil is generally healthy with sufficient organic matter and high CEC (Cation Exchange Capacity) which indicates the ability of the soil to bind with the fertilizers when applied.  Below Optimum Potassium is fairly easy to address with a  separate Potassium fertilizer application.  Let me answer your questions in order so I don't miss anything.  

1. Should you count the prior fertilizer application as part of the recommended amount.? Yes the 28-0-3 fertilizer provides the Nitrogen needed but if you enter the bag numbers from the bag  into the Fertilizer Calculator (found under Tools), the recommended amount for this product is 3.57 lbs. per 1,000 sq. ft.  So clearly if you continued to use this product throughout the year, you would divide up the remainder of the 3.57 lbs. in the applications.

2.  Using 15-23-10 starter fertilizer - Would trying to fulfill the full P & K recommendations without also adding more N be problematic?  Applying a general fertilizer like this will not be a problem for the Nitrogen as long as your turf is irrigated.   Since Nitrogen is mobile (taken up by the turf and going through the soil) the amount available to the turf will vary based on timing of fertilizer application.  To simplify your fertilizer applications, I would recommend you focus on applying a general fertilizer to meet the Nitrogen goal.  The smaller amount of Potassium will be beneficial but not sufficient to significantly affect these nutrients in the soil.  Then separately you can apply a Potassium only fertilizer at a high percentage to address that nutrient.  Most farm mills carry large bags of Potash that farmers use typically with the numbers 0-0-52, 0-0-60 or 0-0-62.  The higher the number the greater percentage of Potassium in the product and the lower amount you would apply.  Whichever product you purchase, come back to the MSU soil test site and enter the bag numbers into the Fertilizer Calculator (again found under Tools) to determine how much of that specific product to apply in a year (pro-rating the quantity for your larger area).  

You will note that I am not recommending applying Phosphorous which readily binds with the soil and is primarily beneficial for root growth for lawns being newly seeded.   A Michigan law recommends not applying Phosphorous unless that is the case and with special consideration about applications around water areas.  For more info, I am including a link to an article about fertilizers and the effect on water.

https://www.canr.msu.edu/resources/fertilizing-home-lawns-to-protect-water-quality

 3. When targeting the K with potash, does it matter whether I use MOP or SOP? Muriate of Potash will have a higher salt content and since it is naturally occurring will be lower priced than Sulfate of Potash (which must be produced).  It you were producing vegetable crops, the difference in the salt may effect the crop production but with turf, you should be fine using MOP at the lower cost.

4.  By bagging the grass clippings for the past 20+ years, we’ve basically been taking nutrients from the ground without ever returning them to it unless we put down fertilizer.  Turf can be very resilient and healthy if you follow some basic principles.  Adding organic matter, providing sufficient moisture and fertilizer are all ways you can thicken the grass to prevent weed seeds from finding soil to germinate.  Specifically:

a. Set your mower at the highest level possible to maintain grass height that will shade the roots during hot temps.  Turf roots are shorter during hot, dry weather and this will help a good deal.

b.  Do what you can to not mow the grass when it is wet.

c.  Make sure your mower blades are sharp so the grass tips are cut rather than torn.  

d. Try to not remove more than 1/3 of the total leaf surface at each mowing (which may require more frequent mowing during rapid growth periods).

e. Increase organic matter by using a mulching kit and return the grass clippings to the lawn.  This would also apply to  mowed leaves in the fall (which may need to be mowed over several times to break them down).

f. Provide your turf with .5 to 1.5 inches of water per week (which would include rainfall and irrigation).  As the article below states, "light, frequent applications of water are more productive than heavy applications once a week".

https://www.canr.msu.edu/uploads/resources/pdfs/irrigation_practices_to_preserve_water_quality_(e0009).pdf

And lastly, I am including a link below to the MSU Extension Gardening in Michigan web site where you will find a wealth of information for homeowners such as yourself.  I encourage you to check out the site and explore the various articles, tip sheets and videos.

https://www.canr.msu.edu/home_gardening/tip_sheets/smart-soils

Good job starting with a soil test to first know the health of your soil.  Going forward, focus on the big picture of helping your lawn be as healthy as possible with the tips and suggestions above.  You should see an improvement over time.  Fertilizer changes some times can take several years so be patient.  

Hope that helps!