I am a huge fan of dice pools and absolutely done with "roll one die vs target". The flat probability you get from one die doesn't give results that feel good.

I was a big fan of the nWoD's S10 system. Add up your stat, skill, and relevant bonuses, roll this many d10s. Every one that comes up as {8, 9, 10} adds to degree of success. Roll another die for every one that came up {10}, possibly repeating if you keep rolling 10s.

You get pretty consistent results. Someone who's a professional will throw ~6 dice on average, so they're very likely to succeed on basic tasks. Much less of that "lol the wizard rolled a 1 and forgot how to read" or "barbarian rolled a 20, I guess he can speak infernal?" weirdness. You still get freak outliers every once in a while, where someone rolls like six 10s in a row and everyone's cheering. But not 5% of the time, and not so binary.

Plus there's other "dice tricks" you can apply for different circumstances. "Reroll all failed dice once", "reroll 9s like 10s", etc.

1d20+stuff is just so basic and threadbare. It's not even easier. nWod's dice pool you don't even have to add. You just count. We all know players that can't add 16+7, but they can probably count to 4.

I don't find "lol 5% of the time something WACKY happens!" very fun very long, no. That is too high a frequency for freak events. Actually, it's 10% because people do wackiness on natural 1s and natural 20s. That's too much! That's so much it's distracting.

I outlined the dice system I liked from nWoD in another comment. You can get some wild outcomes there, but it's not the absurd flat "10% of every roll is insanely good or bad". You get the occasional "I can't believe I rolled three tens convinced the vampire I was a wizard!", still.