Speaking of the Apotheosis of Washington, I’m disappointed nobody (the French revolutionaries?) ever started a republican religion.
All the regular religions are very clearly monarchical. Judaism, Christianity, and Islam all derive from the way ancient people thought about monarch-subject relations. God is our Lord and Master, and we are either His loyal servants or else traitors in rebellion against Him. Prayer is traditionally done kneeling, and resembles petitioning a monarch for aid. Earthly monarchs were clearly thought of as sub-monarchs in God’s larger empire; thus the title King of Kings, which actually reflects the title of the Persian Shahanshah in the sense of “sure, you’re the king of a small subregion, but I rule you as thoroughly as you rule your subjects”. All of this would have been clear and obvious to ancient people who were used to Near Eastern despotic monarchies.
Imagine the sort of religion that would come out of a culture focused on republican virtue, like Rome during the Punic Wars, or Constitution-era America, or revolutionary France. Maybe they would believe in a Heavenly Congress of angels that held spirited debates on whether the harvest would be good this year, or whether they should send a plague. There would be different parties - the Justice Party that wants all human misdeeds to be strictly punished; the Mercy Party that wants to give people another chance. Human actions would tip the balance between the factions and swing close votes on how to treat the material world.
Instead of intercessionary prayer, you would write a letter to your angelic representative, who if they were sufficiently well-disposed to you would bring it to the Congress floor (priests might have special rituals for sending these letters along). Instead of Heaven being the King of Kings, it would be the capital of a sort of cosmic federation, with Earthly republics having the same relationship to it as US states do to Washington. The natural law would take the form of a constitution reserving some powers (the power to set morality, the power to control the weather) for Heaven but delegating others (the power to levy taxes, the power to settle disputes) to Earth.
Most important, the relationship of humans to these angelic representatives wouldn’t be that of abject slaves. It would be that of constituents. The angelic representatives are distinguished, virtuous, high-status people. But they are not your masters and you don’t owe them anything except respect, and furthermore they agree with that. They demand you treat them with dignity, but they would be absolutely horrified if you knelt to them, because we are all in a republic of free and equal citizens and that is not something we do. Human beings have dignity for the same reason the angels have dignity, and they are expected to display civic virtue for the same reason the angels display civic virtue - in order to create a more perfect polity. The procedural rules of the earthly legislature are sacred because they mirror the procedural rules of the Heavenly Congress. Participating in civic life is sacred, because this is what connects us to the angels.
If you live a virtuous life, then once you die you become a voting citizen of the heavenly republic. If you live an unvirtuous life, and bow the knee to tyrants, or betray the republic to corruption and flatterers, then when you die you are denied the heavenly franchise. Maybe you are also tortured forever or something, but being denied the heavenly franchise is the really serious punishment.