"I'm trying to find some thoughts on non-combat dice use. I've tried searching around for posts about non-random resolution but they always seem to focus on combat. I'm curious what people think about the role of rolling in non-combat situations. Is it critical to have those random elements? Critical for what situations?
I know this is very broad. I guess the root of the question I'm trying to answer is why people are opposed to diceless systems. All the answers I have found involve combat resolution."
Sometimes a question is very insightful and strips everything away to reach the core of what is and what isn't.
When is randomness necessary? Why is there opposition to non-random resolution? Knowing the answer to this question and your ability to see the answer is a large part of what being a good Dungeon Master is.
Why do we use a randomizer at the table to determine results?
Why Roll for Resolution?") but that is a little different than that topic. If you're wondering when you should try to resolve something mechanically (whether with dice or with a diceless resolution mechanic) instead of just hand-waving it, that's a good place to start. It covers that topic exhaustively.
Why is randomness popular and why are players opposed to its removal?
Well it's obvious randomness isn't necessary. Games like Amber which uses the players ability to shift paradigms to their advantage and Active Exploits which use the distribution of limited resources to resolve conflict are proof of that.
Dice don't need to be involved to climb a difficult cliff, pick a lock, convince a guard to let the party pass, but an objective mechanic or neutral adjudicator does.
Players are distasteful and wary of the idea of diceless systems because they want to have fun. This doesn't mean diceless systems aren't fun.
Traditionally in life, taking action where you have little insight or control into the results or consequences of that action is a pretty disempowering experience. It's not something people like at the department of motor vehicles and it's not something people like at the internal revenue service, and it's not something people like when they sit down to have fun.
I'm doing a bit of mind reading here, but I'm assuming that "People who are opposed to diceless systems" are opposed to them for those reasons.
This doesn't mean in a diceless system you don't have insight into how things will work out or aren't able to affect or control actions. It certainly doesn't mean your agency is impacted. Quite the opposite. It does indicate that for social games played without a great deal of external structure (say a board with 64 black and white squares) that there is some apprehension that this will be the case.
How to address apprehension of non-random resolution
This really has to do with the individual player. I, as a player, want to insure that I have agency. This means that my intentions when I set out to perform an action are in line with the results. It doesn't mean I have to succeed, just that I feel that the consequences of failure are related to my actual attempt. The issues discussed about agency in the Quantum Ogre series apply just as clearly to diceless systems as to those with dice. Perhaps even more so.
A good place to start would be to discuss the actual mechanic of resolution with your players so they can understand what kind of control they will have and how the game will be fun for them.
In Conclusion. . .
I think the appropriate solution to this is twofold. Examine the reasons why you'd like to make a move to a diceless system. Why a desire for a diceless system? Communicate with your players about why they dislike it. I highly recommend trying a diceless system at least once, it can be quite fun for a short arc or game. And it's an experience you really can't get from a traditional randomized role-playing game.