One of the things that bothers me when I hear "swingy" discussed around D&D is that I don't think when you actually look at where the math goes, it's quite as "swingy" as those throwing the term around want you to think.
I have heard, for example, that "rolling a d20" is "incredibly swingy" because you have one result that is 20 times "better" than another result (rolling a 20 vs rolling a 1). But in general, when you're rolling a d20 (even d20+X) there are usually only four outcomes:
1. Critical Failure (in 5e, this is rolling a natural 1; in Pathfinder, this is "rolling 10 less than the target number")
2. Normal Failure (in 5e, this is rolling something less than the target number that is not a natural 1; in Pathfinder, this is rolling between 1 and 9 less than the target number)
3. Normal Success (in 5e, this is rolling a least the target number but not a natural 20; in Pathfinder, this is rolling between the target number and the target number plus 9)
4. Critical Success (in 5e, this is a natural 20; in Pathfinder, this is rolling at least 10 greater than the target number - and in some cases for various editions of D&D this could encompass additional numbers - e.g., 19-20 got you crits for an axe in 3e).
Usually these spreads look something like 5% vs 40% vs 55% vs 5% in 5e, though if you're fighting something with a higher armor class, this might adjust the other way. But in general, the outcome isn't 20 times better, it's really a "hit or miss" and things aren't really very "swingy" since the probabilities wind up being approximately equal (or, for "because it isn't fun to miss" usually stilted a bit towards "hit").
Damage dice is where things
are swingier; however, damage dice are frequently smaller than a d20 and paired with a bonus; for example, a weapon that deals 1d8+3 (thanks to a high strength score) has a range of between 4-11; while rolling an "8" on the base die is less than three times as "good" as rolling a "1." As the +X goes up, the swinginess diminishes. Similarly, as the number of dice goes up, the swinginess diminishes (as you go from a linear distribution of outcomes to a bell curve; getting a "2" vs a "16" on 2d8 is possible, but those are statistical outliers - most of the time you'll be getting a result of 7-11, which does have some swing to it but is far less extreme).
Again, the swing is noticeable at low levels where hit points and bonuses are small so one high damage roll might be able to take a character out of commission in one shot and it happens frequently enough to be noticeable (say, 1 in 8), but quickly becomes less as characters reach 3rd-5th level where it's going to take a lot of dice all coming up high at the same time to take a character out in one shot (say, the odds of hitting 24 damage on 3d8 damage dice is 1 in 256) and is extraordinarily unlikely at high levels of play.
This is not to say I'm not interested in how other systems might do innovative things with dice (e.g., Force dice in Star Wars) but I'm a little wary of claims that "D&D's design sucks because it's swingy" - it's not as swingy as you might think.
And yes, I'm all for letting the DM letting the dice fall where they may... with the caveat that if a DM drops in an encounter and realizes within a round or two it's clear this is going to wreck the PCs because the DM overtuned the encounter (e.g., even a "below average" hit from the baddie one-shots the toughest PC when the intent was a challenge that would tax them but not kill them), I have no problem with the DM dialing things back a bit on the fly; however, if things go poorly because the PCs have a string of bad die rolls or the DM rolls three natural 20's in a row for the monster... well, tough. Them's the breaks. Some uncertainty ("swinginess") is good. It also allows PCs to pull of surprise stunts that totally throw off the DM's plot but that the DM should also be fully prepared to accept and roll with (obligatory Critical Role/Dust of Deliciousness nod).
