all divine magic is bargain magic with powerful beings
That's not really true, or at least it hasn't been for at least two editions (3e and 5e; 4e was, as usual, Different™).
Divine magic in 3rd and 5th edition is magic that comes from commitment to...
something. But there needn't be any "person" or even "cosmic force" on the other end. Divine magic can literally just be "I think Joy and Hope are
so bloody important that I can literally make miracles happen because of those thoughts." Where the Wizard (and other "arcane" magic users) in some sense trick or force reality to do as they will, the divine magic-user instead basically just has such absolute, ironclad
conviction that reality just goes along with it (up to a certain point.)
Now, it is often
easier to access this sort of thing through a powerful divine entity, such as a deity, or a potent cosmic force, such as the Light from the WoW setting, or the Force from Star Wars. But at least 3e and 5e both explicitly allow you to have divine magic come from some other thing instead.
4e, as noted, did not do this--but that's because (a) the general function, aka Role, provided by Clerics was
explicitly also provided by many other classes with no divine flavor at all (Warlord, Bard, Ardent, Artificer, and Shaman depending on how you view the Primal Spirits stuff), and (b) they actually gave a
reason why the Divine power source works this way in-story....and that reason happens to NOT actually
require faith, believe it or not. So, in one sense, your question is answered for 4e by saying "sure, there are at least four other classes that do what you want and have nothing to do with any particular belief." In another sense, the answer is, "they gave a reason, and that reason ALSO doesn't actually require what you're talking about." (Specifically, the Divine power source comes from having received Investiture from a being capable of doing so--usually a god, but sometimes other things--but that transfer of power is pretty much
permanent, so it's entirely possible for a deity to give Investiture to someone that DOESN'T believe in them, and likewise, possible for someone to lose faith and leave without having their powers taken away).