I've been messing around with randomness for a long time. There's this whole sub-sub forum called "Roll To Dodge" which has games with randomness down to the core which I frequent. And I've played plenty of Roguelikes and recently Civ4. Those are where I gained the pessimistic attitude. They all have quite a lot of randomness.
Actually, I've written some pretty extensive rants on the topic of randomness in games before. (Surprising, I know.)
There are ways to use randomness that are better or worse when it comes to specific styles of play. In some games, what I would call "Candyland-Type", you have essentially no control over your destiny because the RNG is the ultimate arbiter of your destiny. You don't really have much strategy in a game of Monopoly, you just have to hope you land on the best properties first, and never land on Boardwalk with hotels. There's no point in thinking or planning in a game like this, as you have no power or control. It is fun only for those (usually very young) players who want to win without skill, as completely random games will let anyone win an even number of the time. A Chess-type game, however, has no randomness aside from what the players are thinking. These games favor forethought above all else, as without randomness, you can plot out and predict the permutations and consequences of every move to the limits of your mental capacity.
With that said, there are two very different types of mixed-randomness games - Card-type, and Dice-type.
Dice-type games let you choose optimal strategies and try to stack the odds into your favor, but it's ultimately up to luck whether you succeed or not, and there's little point in planning more than a few moves that rely upon luck ahead, as you have almost no clue what the results of any given action will be. These are, notably, most tabletop RPGs and games that rely upon dice. They discourage planning, and enjoy making everything rely upon the fact that at any moment, a failed saving throw can make the best plans completely moot, and the dumbest lucky actor a winner. They tend to frustrate players who like to play strategically, because it makes strategy fairly pointless if whether or not you can even move to the position you need to be a matter of dumb luck.
Card-type randomness, meanwhile, often forces random events onto you to which you must react. This means that you have unpredictable events happening to you, but where the results of
your actions will always have predictable results. Games like the original Avalon Hill Civilization board game (IMHO, Sid Meyer's Civilization is not as good a game) will have you expand your populations, move them, build your cities, and then randomly give you trade or disaster cards, which you can then trade with and try to resolve those disasters in the manner that harms you the least. You can expect, in an average turn, to lose three cities a turn when it really gets going, so you can always just make yourself prepared to build three or four cities the next turn to compensate.
Card-type randomness is often superior because it injects a randomness that makes the game less utterly predictable, but at the same time, makes every player choice actually matter and makes them fully in control of their destiny, even as random events are forced upon them.
DF has the fortunate tendency to often have card-type randomness (Except for, unfortunately, combat, although you have so little choice in combat to begin with that it's almost Candyland. It's part of why I dislike much of the combat in this game.) and in the case of mining, just consider that you weren't guaranteed those minerals you mined beforehand, either.
Until you actually revealed that mineral you wanted, you didn't know it was there. It could have been more solid granite wall, for all you knew. It was a card-type randomness that you actually drew the mineral you want in the first place, and now that you have the chance to draw a boulder of that mineral, it's really like drawing a card from a special deck. Sure, it may not work out, but you are still in the random phase. When you actually eventually do get a lucky roll, there are no other forms of randomness forced upon you.
Unlike Sid Meyer's Civilization, there's no chance your battleship is going to get sunk by a barbarian bowman. What you do with the minerals when you get it have little randomness (excepting the dice-type randomness of quality, although that is rarely all that important)