The Tavern

The core proposal is to add a good mutation that improves the rate of piety gain like wearing an amulet of faith. Beyond that:

I propose having the mutation deprecate actual amulets of faith and allow the inert amulets to be removed without penalty.

I propose granting the mutation as an intrinsic to Hill Orcs, possibly coupled with a decrease in Beogh's natural rate of piety gain, and earlier Temple generation for Hill Orcs.

I propose adding a bad mutation version that reduces the rate of piety gain, which would cancel out the good mutation when both are obtained.

Why? Mainly to add some new interactions for replay value, particularly for Hill Orcs whose only interesting feature right now is that they can worship Beogh. This would make picking a Hill Orc a more attractive choice for players that aren't planning to worship Beogh. (Right now, when playing a Hill Orc, worshipping Beogh is almost a no-brainer.)

Crawl used to have a -faith mutation called 'forlorn', and it turned out it was a bad idea.

With respect to HO, I don't really agree with your premises. They are very strong with many gods, because they already have an extremely high invocation aptitude (tied with DD/Ds) and of course because they're a strong species overall; I don't think choosing beogh is at all a no-brainer, especially if you aren't fond of ally management. Buffing the HO/religion interaction even more seems unnecessary.

Oh right, I remember forlorn now that you mention it. I don't know why it was deemed a bad idea, though if I had to guess I'd say either it interfered with piety-driven mutation gain from Jiyva or that it just bothered people who got hit by Malmutate (but then, what doesn't?). I am a bit curious if it would still be considered problematic when paired with a good mutation counterpart and in the post-mutation-potion-rework environment, but in any case, I wouldn't mind if it weren't brought back.

I guess if you want to completely avoid spellcasting and make heavy use of Zin's abilities, Hill Orc would be the best choice, but that's a rather niche case. Maybe I'm just unusually attracted to interesting species, but whenever I've wanted an Invocations-based character who doesn't worship Beogh, I've always gone for Deep Dwarves, Demonspawn, or Formicids. Then again, I've never once wanted to play a Human, so maybe they would be a better choice to get the pious mutation.