Wouldn't that explanation only work if their load is high enough that they frequently hit the limit on how much can be shipped per day, but they rarely hit the limit on how much can be shipped per 5 days?

If the load is lower than this, then deferral of non-Prime orders would rarely be needed. If the load is higher than this, deferral would rarely help.

If their load is in that window where deferral can help with load, the amount of deferral for a given warehouse should vary from order to order all the way from no delay to several days.

My recollection from before I got Prime was that almost all my free shipping 5 day orders shipped the same day, and then over a very short time that changed to them almost all being delayed by an amount that depended on how far the warehouse was from me. I lived in the same city as their biggest warehouse, for instance, and orders served from there were almost always delayed 4 days and then overnighted. That indicates that the deferral was not based on load.