In reference to your statement, "Why in the world would a stay at home order effect the mortality rate?".
So, in a given 1000 people, if we can decrease the number of those who are high risk for dying getting it at all, it absolutely would decrease the number of deaths in that 1000.
If we are/were able (through a stay at home order, mass education, or whatever other means) to make the infected part of the population look more like group 2 than group 1, it can absolutely impact the percentage of those infected that would die. There is no way that group 2 (particularly if we added a few zeros to my example) would have the same rate as group 1.
For instance, I work in a large healthcare system and when this all started they almost immediately pushed for older employees or those that self-identified having respiratory or conditions that impact the immune system to be able to work from home and self-quarantine. Additionally, there were a number of people who knew they were higher risk that chose to stay home, rather than work (I think of a courier I interact with daily that is 85 y/o who chose to use a ton of PTO). These things do not happen during a regular flu season and (at least in my case) wouldn't have happened without a shutdown of portions of our work. If those same type of people had been out and got the virus, it would have increased the percentage/rate of infected who died. Obviously, we will never know how much.