Adam, Thank you for your question and I'll try to answer as simply as possible. While the addition of the substances you mentioned will help improve the nutritional quality of poor soil especially if it lacks calcium, they will do little to nothing to reduce compaction of clay soils. The best practice, unless you want to till your lawn and start over, is to continue to core aerate your lawn each fall and spread organic matter such as compost, finely ground mulch or even ground up leaves and let it work itself into the aerated openings over the fall and winter. Even if you choose the "start over" option, the remedy is still to mix lots of organic matter into the existing clay soil.  The organic matter improves the soil by adding air space into the soil which is missing in heavy clay compacted soil. Here is a link to a good fact sheet from the University of Maryland Extension on improving soil health and improving compacted soil: https://extension.umd.edu/resource/soil-health-drainage-and-improving-soil.
   If you continue to use the core aeration and add organic matter each fall, you will slowly but surely see an improvement in your lawn compaction problem.
   Thanks again for the question and have a great spring and summer.