As I've been looking through a couple games lately and playing some 5e, I've noticed that the psychological stats are a bit lackluster or hard to use. I think the GUMSHOE system gets stuff pretty close to right, however.
What are these "psychological stats"? In traditional D&D, you have alignment. In the World of Darkness games, you select a nature and demeanor (mostly). 5th Edition D&D has Ideals, Flaws, Bonds, and Personality traits. DungeonWorld has bonds and alignment. Fading Suns actually has opposed traits (ego/faith, passion/calm, etc.) which kinda do some of this. But these are all basically little descriptions of how your character is supposed to act. The problem I see, is they don't always accomplish much.
Older edition D&D alignment basically let the DM punish characters who acted out of character. World of Darkness games let you regain willpower by acting in accordance with your nature, which is a bit nice, though demeanor generally did nothing. 5th Edition D&D has the inspiration system which gives you a tiny bonus if the DM agrees you're acting in accordance with your psych stats, but my current GM hasn't overtly used it since the first or second game. 13th Age doesn't even really have a psych stat at all.
What I really like, however, are the drives of GUMSHOE (e.g. Trail of Cthulhu) and passions of ORE (e.g. Reign). With ORE, you have three passions but you only have one thing that drives you to investigate in GUMSHOE. You could easily adapt this to your drive to adventure in a more traditional fantasy game. With ORE, when you act in accordance with one (or more) of your three passions, you can claim a 1 die bonus per passion, or the GM can levy a similar penalty if you're acting against them. Your drive in GUMSHOE can similarly affect your stability whether you're acting in accordance with it or not. GUMSHOE also suggests drives be used to edge characters into the adventure.
Why do these seem like better systems? For one, its a relatively small bonus and easy to track. I think GUMSHOE does a little better than ORE for simplicity, but they're fairly similar. The great thing is you can keep applying this minor bonus constantly, unlike inspiration in D&D 5th. Advantage is a small bonus when applied sparingly, but its rather huge if it were constantly in effect. So a drive or passion bonus in a d20 type game would need to be something smaller, like just a +1. An issue there is these little bonuses nickel and dime the game towards a big mathfest. I'm curious if a "lesser advantage" system of rolling d20+d12 choose the highest would actually be any advantage or not. Mathing it out looks like its about a +1 since your average result is higher with this than just a d20. However, given that a roll of 10-12 is still going to be a failure, I'm not sure if it'd really be enough of a small, constant bonus. You could do something similar with a d4 in Dungeonworld: instead of 2d6 for your roll, it'd be 2d6+1d4 keep the two highest. You'd be a little less likely to fail, though not much more likely to get that grand high roll. Dunno if that really maths out well, but people like rolling extra dice. Another option with DungeonWorld is changing the system to 2d8 (9+ is ok, 13+ is exceptional) then a reliable +1 isn't quite as big as it is in the normal system.
Advantage in D&D 5th basically puts you at the mercy of your DM as to your good roleplaying. Passions or drives, on the other hand, are easier to sum up and you can remind your GM that you might merit the bonus: that seems crass in D&D 5th and possibly in some other systems. So the size of the bonus matters (gotta be small so it can be more common) but also the method of bringing your psych stat into play.
An alternate bonus would be awarding XP. That doesn't work in modern D&D when characters just end up leveling at the whim of the DM. In 13th Age you could at least hand out an incremental advance, I suppose. In a game like WoD or Fading Suns where XP works differently, you can easily give XP for adhering to your nature or suffering from your flaws. TSR era D&D could certainly get a small XP bonus (+5%?) for adhering to your alignment each session which might be enough for motivation and to inspire some roleplaying.
I'm not sure that psych stats would necessarily fix everything, but if they were better implemented in some systems they just might help a bit. Adding a Drive to DungeonWorld or D&D (old or new) wouldn't be too difficult if it had almost no mechanical effects, but it seems that the real benefit of these psych stats is to help people play their character and give them a benefit for doing so.
There's an ancillary issue here too, which is constraining choices to fit a group. I honestly liked the constraint from the Temple of Elemental Evil vidya game, which was you picked a party alignment and then created characters. Back when alignment at least constrained some classes, it meant Paladins could only be in a LG, NG, or LN party. Barbarians couldn't be in a Lawful party, while Monks couldn't be in a chaotic party. Binning drives into similar themes might not be necessary, but its worth thinking about building the party as well as the characters.