Really want to know why gypsum doesn’t work here? In arid parts of the country, sodium occupies many of the cation exchange sites in the soil. And since it is only a +1 charge, soil colloids tend to disperse and can be easily compacted together causing a poor soil structure. Adding gypsum (CaSO4) allows the Ca++ to release and replace the soil-bound Na+. The released Na+ is leached out as Na2SO4, and the soil tends to granulate due to flocculation (fluffing up and colloidally glued together on the microscopic level) with more Ca++ on the exchange sites. This granulated condition improves soil structure, and soil is then less prone to compaction. Since there is far more calcium than sodium on exchange sites in our Louisiana soils, adding calcium via gypsum has little or no effect on most of our soils.
Instead of gypsum, consider core aerifying in spring, summer or early fall to reduce the compaction and improve plant health. Aerifying with large half-inch hollow tines and punching about 25 holes/square foot will produce good results. Very compacted soil can benefit from several corings each year (common with sport fields). This does not minimize the benefit of have a high Ca and Mg level reading in your soil test, but coring does physically reverse a physical compaction of soil particles. Yes, coring is an expensive service to buy, and most lawns really don’t need it; but it won’t hurt. And if the soil is compacted, it’s usually the best solution to the stifled growth.