The Price is Right Core Mechanic
I started to like 1d20 roll under for a couple of reasons.
It keeps the maths simple, with 5% increments that feel meaningful but aren't as fiddly as d%. By using an over/under system; rolling under your stat but over a difficulty rating of 5 or 10 (with exact numbers being a crit success) we end up with a core mechanic that is pretty direct and you can pack most of the stuff players need to make decisions right there on the sheet.
There are, of course, quirks to this.
Firstly you do lose natural 20 as a critical. The odds here have not changed. However Nat 20 has a certain meme quality that shapes thinking, and the casual player values familiarity over simplicity.
As such you will watch them roll high, look at the dice and then look deflated as they realise this isn't D&D.
In a similar vein rolling low when the task is difficult allows you to watch the arc of someone realising their roll is under their stat, but too low. Like you've stolen their success from them.
What players want is to know if they're rolling high or low. They don't want to be told it's like The Price is Right. Which is annoying, because this system should work. It does work.
You could simply calculate character attributes and then give them a -5 (difficult) and -10 (hard) rating. This would allow you to keep everything on the sheet, but you do end up with a table and some players do get lost the moment they see that many numbers in one place.
To retain roll high you would either need to invert attribute rolls or, more sensibly, use something like a standard array. Now you need to roll over your attribute score, lower attributes are better, which will also feel wrong to players. Minus numbers being good and positives being bad feels similarly wrong.
You could, of course, apply those numbers to the roll instead of the attribute. However I like my mechanics to have an arc where any mathematics and bargaining takes place before the roll, and once the roll is pitched it's a straight line to the outcome.
Besides, once you're modifying the roll we're no longer working off a number on the players sheet, which is really what I want.
So The Price is Right takes it; it does enough of what I want that I don't mind overlooking its foibles.
That and, at the end of the day, I need to write for the players I have and not some mythical D&D player who won't pick up my teeny-tiny indie game anyway.