As your clarifications to the question suggest you realize, the problem is not Rule Zero; it's your group wanting to change the rules to make the game easier for them, and you not wanting to. That's certainly understandable, but try looking at it this way: Your players are telling you something important about the kind of game they want to play. Specifically, they're telling you that they really, really don't enjoy losing.
Pandemic, with the kind of changes you mentioned above, leaves very little chance of failure. It's not so much a "game" with risks (Will we save the world or not?) as it is an exercise or puzzle (How can we go around cleaning up these diseases? What would the sequence of events look like? If a move has bad consequences, let's undo it and see if there's a better way.) So if you want to have fun playing RPGs together, I think you'll have to make sure it's that kind of game. There are a couple different ways to do this while still sticking with your goal of following the written rules given.
Picking a game
You have a big advantage here in that you're just beginning the hypothetical process of picking out a game. I can't give you a laundry list of recommendations, but I would strongly suggest going in one of two directions:
- A game where you are expected to fail, and fail often, and that's all part of the fun. Paranoia is one example of this. It's probably more competitive than cooperative, and you start with backup clones, which you're expected to go through several of as you die several times a session. It's not so much about succeeding as it is failing as hilariously as possible.
- A game where there there's really no mechanism for failure. Microscope is a great example: there's no GM at all; the players take turns dictating that something happens, and with few exceptions, that's now what happened. If another player doesn't like it, you can just create new story points before or after that point and not think about it too much. There's literally nothing that could be called success or failure; at the end of the session, you've got a story. At worst it might be more satisfying to some than others.
What these two games have in common is that they're not really so much about winning or losing; characters can die, but they're not associated as strongly with the players, and the point is to have a good time building a story together. You can ask in a forum or in chat for more examples; searching for humorous games and "story games" respectively might be a good start.
Picking a game style
If you want to stick with a more "normal" RPG like D&D, however, it will be important to establish an atmosphere early on. What some people forget is that D&D is not a game; it is a system of rules, which you can use to create a game. Each individual game will have different "settings". The Same Page Tool is one way to think about the things that the rulebook doesn't lay out for you, like how cooperative or deadly the game is going to be. These are decided in every game, consciously or unconsciously; in your case I'd say it's important to be deliberate, especially if your players have heard that D&D is a game of the DM against the players. That's true only if you want it to be; in my games I usually explain to the players that I see us as partners in creating a story about overcoming obstacles, and while I'm more responsible for the obstacles and they're more responsible for the overcoming, I'm rooting for them the whole time.
Here's one example of what that looks like in practice. The key point here is this one:
[This game] uses “cinematic” realism, where setbacks will put you further away from your goals, but it’s expected that you’ll achieve them eventually.
In other words, in that particular game I went so far as to say I'm not really going to let you die, unless you tell me you want to do so dramatically.
The core issue here is trust. You have to explain to the players that you're not going to kill them; they have to believe you; and then you have to stick to your promise (and depending on their experience and attitude, you might have to go out of your way not to kill them, even accidentally.) How to establish that trust is another question, but at least it should grow with time.
Note that none of what I've said so far necessarily involves changing any rules. It's certainly easier to keep the PC's alive if you're willing to fudge the occasional die roll, but you can also do it just by exercising the control you have over what there is in the world, how it responds to the PC's, and how it behaves if it takes a dislike to them. The GURPS 4e Basic Set has a whole section on Keeping the Characters Alive on p. 496, including this gem:
Not every encounter turns hostile; not every hostile encounter turns violent; not every violent encounter involves weapons.
In particular, there are very few things most creatures, sentient or otherwise, are willing to fight to the death over, particularly if it starts to look like it could possibly be their death.
Once your players understand that you're not out to get them, and they're free to explore, they'll probably cool it with the requests a little - not because the game isn't easier than an average RPG, but because you're in charge of making it so. You, as the GM, have the lion's share of the responsibility of creating a fun experience, adjusting or following the rules in whatever way seems best to you to accomplish that goal. That's what Rule Zero is really about.
A couple of final thoughts:
- It's always good to talk to your group and ask what everyone wants. I hope my ideas are a good starting point, but firsthand info is much more reliable than the musings of some Internet stranger.
- If one of the options I've described doesn't sound fun for you, then either don't play it or don't volunteer to GM it. You are also one of the people who needs to have fun here, and if you're all about the rules (called "nomothetic") and everyone else is all about changing them, it can be a bad time (though that's an extreme example). Keep in mind that, as mxyzplk and WrongOnTheInternet point out, no standard game has rules for every situation, and you'll probably encounter some where the rules as written lead to weird results, so you're almost certain to have to make or change rules sooner or later. (Indie games are more likely to "cover everything" by having very broad rules for how the story goes and none for action resolution, but you may find that unsatisfying as well.)
- I can empathize with your players a bit. Though I wouldn't carry it
that far, I too have been known to think "life is tough, when I'm
gaming to relax I want to win, not die." You or your players may find the answers to
my question How can I get less attached to my
characters? helpful; I know I did.
- Start small. While I think it's worthwhile to try to anticipate problems in time to avoid them, mxyzplk is correct that you and your players will develop your attitudes and play styles over time, so it's best to not commit to a many-week endeavor without some idea if you'll like it or not. Microscope, mentioned above, can be introduced and run in 3 hours (though 6 may be more satisfying stopping point if people are enjoying it; there's no end as such.) Some games are specifically designed to be even shorter, and even complex ones like DND can be taught and run in a single session with careful management.
- By the way, having beaten Pandemic a few times on
introductory difficulty, my group now plays on the hardest setting,
sticking with the rules. Once they feel safe enough, maybe yours will grow
to appreciate a challenge as well :)