I think there’s a bit of modern bias in this tangent. The ancient Romans, for example, didn’t care what a person believed, only that the performed the right rituals at the right time, orthopraxy (right practice). In modern times we center having the right beliefs, orthodoxy. This is a split in polytheism, which favors orthopraxy, and monotheism, which favors orthodoxy.
From what I've read about ancient Greek and Roman polytheism, you are right, but one's religious life went far beyond state ritual. I will, however, admit a modern bias in terms approaching fantasy polytheism based on my experiences with modern religions. Of course, we are talking about fantasy worlds and fantasy religions. Further, it is a game and there are mechanics in the game related to the practice of religion. A certain amount of simplification and hand waving is expected.
Applying that to a truly polytheistic world like your typical D&D setting and all you’d need to do to switch deities is pick up the rituals and practices of your new god. Though the specifics of the mortal organization accepting you would be a different matter entirely.
Yes. Except that the mechanics of divine magic in the game imply a more personal relationship with the god. I don't have an issue with the PC acting outside of a religious organization. I kinda think of most D&D cleric and paladin PCs -- well, really, most PCs of any class in the play I typically encounter--as being iconoclasts. As a DM, I generally let the player decide what their character's relationship with the divine and with religious organizations is and try to work that into the campaign.
But then that would seem to point to the person themself having the power and the deity being the conduit. Can you be a cleric or paladin devoted to an ideal instead of a religion? Why not?
Sure, if that is how the player wants to play it. In D&D, however, I typically see a world-building assumption that various ideals are within the domain of one or more gods. Would those gods grant you powers just for their devotion to the ideal and not a particular god--depends on the world building and play preferences of the table.
I've approached all of this in different ways in different campaigns. In most campaigns, I find players playing clerics and paladins tend to roleplay devotion to a specific deity, if they even roleplay it at all.