Sounds like meddling gods is a thing for you!
Among other things. My take on this is inspired in large part by Greek and Hindu myth. Odysseus is always getting messed with by the gods. Intervening on the PCs behalf is the positive side of that, but it gives me an excuse for putting the characters in epic situations.
Another important aspect of this which you haven't observed is that I took the choice out of my hands and gave the player a fortune roll to resolve the outcome. I still have some leeway to meddle, but if you come up snakes eyes (or some other low score) then no divine intervention for you - your god wasn't looking in your direction, your god didnt' think the request was worthy, or whatever. By dicing for the outcome, even if the chance of sucess is remote (most of the time, you need to roll 'box cars' to get divine aid even in favorable circumstances), I can maintain my nuetrality as a referee.
I also feel I have to digress a little bit into real world religion and its applicability to D&D campaign setting. Most people's take on D&D religion is what I call 'post-Christian'; that is, it is informed by a generic Judeo-Christian world view, but it in fact doesn't agree with actual Christian theology. It's a superficial take on it that you might have if you'd been to Sunday school when you were six or by osmosis from the general culture, but weren't yourself pious. Of course, you've every right to impose on the fantasy world any take on religion you prefer - generic post Judeo-Christian is as 'right' as Greek or Hindu in a sense - but from my part it does involve imposing some ideas that seem really odd to me in the context. Some examples:
1) Clerics are powered by their 'faith' or divine magic is powered by 'belief'. Leaving aside that this is probably not the sense that Faith is used in actual Christian theology, it seems a strange thing to impose on a polytheistic world particularly when it is in stark contrast with the mechanics of the game. Clerics don't need faith in D&D. The have no faith scores. They cast spells which are granted to them by divine powers quite apart from any faith. They arent' miracle workers, but spell casters. Indeed, most real world religious systems do not stress anything about the importance of faith, and instead put the emphasis on ritual and regulation and what are in effect economic transactions with the gods - sacrifices made in exchange for divine favor. In a sense, the shallow sort of definition of 'faith' in common usage today - "Does God exist?" - is not only not a Christian concept, but seems wholly irrelevant to and unsuited to most D&D worlds with their spellcasting priesthoods.
2) One of the strangest things to me is the notion that the Cleric is some sort of 'D&Dism'. In truth, it's the Wizard that is the D&Dism. If you look at real world magical systems, almost all of them would in D&D fall into the sphere of divine magic. Almost all real world 'wizards' up until very near the end of the 19th century when magical systems based on science (or rather psuedo-science) started appearing are best classified in D&D terms as clerics. Almost most no historical magician believed that he himself works the magic, but instead believes that he influences some divine power or spirit into performing magic on his behalf. Almost no historical magical tradition believed it could perform the sort of flashy magic most closely associated with the D&D Wizard. Instead, the spell list would look a lot like some subset of the clerical list. If I was trying to recreate the world of antiquity as the ancients beleived it worked, it would be the Wizard I'd ban as a class, and all 'wizards' would be clerics of some sort.
3) In the D&D sense, Christianity has no clerics. The last Christian clerical tradition in the D&D sense died out with the Catholic counter-reformation in the 16th century. Christianity used to have Theurgists who cast spells and this used to be a somewhat approved method of interacting with the divine, but no longer. Instead of spellcasters, Christianity only believes in divine intervention now - miracles worked on behalf of the faithful often regardless of their clerigical caste.
4) In polytheism, it's really unusual to only worship a single diety nor is worshiping a variaty of gods generally seen requiring you to conform to a particular one or a betrayal of others. Yet it is a truism of D&D that just about every character has a single patron diety. It's polytheism, but its a polytheism where every diety is like a jealous Judeo-Christian God.
5) Christianity postulates a God which though active prefers not to be seen in the world except by the faithful. We could argue elsewhere over whether that is because such postulate is necessary in a world where God doesn't actually exist, or whether its because God is actually nicer than the meddling Greek gods with their acts of hubris and displays of power and wrath, but which ever our take it seems really strange to imagine a D&D setting with a comparitively remote pantheon of Gods given that every 30th person seems to be a divine spellcaster getting spells from some divine intermediary and very much openly messing in the course of history. We've got a bunch of post Judeo-Christian gods of still small voices who give every village priest in a dirt floored chapel spells with which to boom their power.
I bring this up, because EnWorld's prohibition against real world religious statements often seems to me to be broached by statements of D&D "theology" that intersect with points like the above in cases where the writer is voicing unreflected on "common sense".... that actually has a basis in real world religion. It's hard for me to not bring up real world religion, when I get in to threads where everyone is swimming in a broth of unstated but assumed religious notions. This often occurs for me in threads that aren't as overtly on religious topics as this one. For example, threads where the poster wants to ban clerics as 'unrealistic', it's hard for me to avoid the fact that the poster believes that only because the religion he has a passing familiarity with banned D&D style clerics, and the Wizard he's familiar with from video games and the like is actually the D&D wizard and not Cabalism and Hermeticism (not that that is entirely a bad thing).
Anyway, I don't actually want to argue anything about real world religion, I'm just getting tired of not being able to point to the gorilla in the room, which is namely, D&D religion is informed by what people assume about how real world religion works and not anything actually found in the setting or the rules. As some pointed out, 1e D&D explicitly had a table and a fortune mechanic for resolving the very point of what happens when a character appeals for divine aid. The only reason it should be surprising that D&D deities are active and willing to intervene is because the person assumes that's the way it works in the real world, regardless of what the text says.
I typically prefer a game where the gods are more distant. Meeting a god is like streets ahead of meeting the emperor, which is greater than meeting the king, which is greater than meeting the local baron, which is greater than meeting the mayor of your village, which is maybe something 1st level characters can do, if he's not too busy.
Actually meeting a deity physically is a much rarer thing. Those that do meet with dieties face to face on friendly terms are called Saints and are generally venerated. One of the characters in the campaign I'm currently running has acquired the reputation of a Saint which ironicly, isn't true. The closest he's come to his diety is through dreams and visions and the usual excercise of divine spells. But recieving the active work of a diety is something a peasant might expect at some point. And suffering from some god's divine wrath against the community is also something most peasants might reasonably expect at some point.
That might change, depending on the setting's other trappings, but generally if the mayor ain't gonna chip in to save somebody, sure as shoot a GOD ain't gonna do it.
My mayors are generally more active than usual as well, but there is a portion of your analogy that just doesn't work for me. Most of my mayors are like 50 year old 5th level Experts which no physical scores (STR, DEX, CON) above 9, no combat related feats, no combat related skills, often the Non-Combatant and Civilized traits, and no magical items related to offensive action. My mayors are generally comparitively to D&D tropes quite willing to help PC's in every way that they can provided the PC's present themselves as honorable persons, but really, what can such a character really do? Even when the PC are 1st level, they often have greater martial prowess than such a character. Beyond that, the mayors don't enjoy comparitive omniscience and omnipresence. They can't (for example) by concentrating see everything within five miles of every lit fire (per for example the 3e Dieties and Demigods), can't just pop in and help (per every editions Deities and Demigods), but must sludge through whatever obstacles that the PC's are facing to get to where they could help them, and mayors generally command no more than a dozen armed magistrates rather than legions of spiritual beings. It's not that the PC's haven't occassional recieved aid from such groups, because they have, but the aid they can offer is small and generally results in dead NPCs. Ultimately asking such NPCs for aid generally involves sacrificing NPCs to save the PCs, rather than the PC's shielding and protecting the NPCs. The same is not true of asking aid from the Gods. Arguably, many of the gods involved in the current campaign would rather the PCs ask them for aid than they ask the mayor.
