But, on Beluaterra and Dwilight at least, there are mystical and magical powers, and they are not a vanishingly small part of gameplay.
You have missed the point.
First of all, no player characters have magic. Period. (Scrolls don't count; they're created by NPC magic, but don't require magic in the person casting them—that's their whole point.)
Second of all, to give such powers to religions would be to legitimize
all their Gods. And giving it to the ones who don't even claim that there are Gods—or that the Gods don't interfere in mortal affairs—would just be bizarre.
Even aside from that– here's a half-dozen possible non-mystical changes that make sense and would help:
Make religion members able to give flag something as "heresy," like vulgarity, and it automatically goes to the religion's elders, who rule on it anonymously. i.e. "the lesser nobles are grumbling that you've started to stray from the One True Path"
OK, that doesn't sound unreasonable. What sort of effect would you say this should have?
Stop making your own realm's troops throw priests in enemy dungeons.
Not sure what you mean here. Do you mean "change the code so that priests of your own realm cannot be arrested"?
Create a mechanism for an RTO where a priest can make someone else the lord of the region, besides themselves.
Again, it's not an RTO. It's claiming the region for the religion. I could see a case for making someone else who's an elder the Lord, but that's pathetically easy to game.
Allow religions to declare realms "evil" or "misguided" or "faithful" or "ignore."
That sounds reasonable to me, but I think Tom would dislike it, as it mixes the religious side and the political side too directly.
Allow religions to declare characters anathema, whereby they may have a random chance of being assaulted by peasants in areas where that realm is dominant, especially if they're a courtier or something. Obviously the number of anathema persons should be capped, probably as a ratio of the members or priests of a religion.
That one sounds kind of cool. I wouldn't make the chance dependent on class, but rather on unit size, though.
Boost the H/P available in the priest game.
Done. I have just tripled the amount of H/P available to priests
</snark>
...Yeah, there probably should be more H/P available to priests. Maybe for converting large numbers of peasants and the like.
These all seem feasible. These all are non-mystical. None of these have been previousy explicitly rejected. None of them give religions some kind of god-moding power. None of them could substantively replace RP and influencing by players. All of them could plausibly augment and encourage such– and even the most peaceful and get-along religions will eventually be tempted to declare someone anathema or proclaim a realm evil.
The only point I'd make in response to this is that most of these can be done in RP. Obviously that can't apply game-mechanic effects, but it can definitely serve the purpose of, say, getting the characters in the religion to view realm X as evil, or Allison Kabrinski as a heretic.