I really like this idea, it would make it so we don't need so many scaling DCs to prevent auto-critical on checks from a few levels ago, and needing to roll a Nat 20 to pass something that would be trivial for a higher level character.
But I believe it might have a couple issues if we are too varied with the ranges. For example, a Player wants to do something unexpected, DM finds or sets the DC at 16. In the current system if the player rolls a 26 or a 6 the DM knows if they've Botched or Crit, and if the DM told the player the DC they also know how they did, so fewer hard feelings.
Issue being if crits can be anywhere from +1 to +10 the DM then has to think what a +/-4 critical is worth, then on top of that what hitting +8 means. And then the player rolling a 23 might think they crit when the DM set the crit range at +/-8s, which might make the player feel slightly cheated.
A way around this would be to standardize what types of roll crit with which numbers. For example, keeping Attack rolls as +/-10, then change skill check to +/-5, then for an epic campaign roll like a unique ritual every +/-2 meaning something horrible or wonderful. I think it would help reduce mental gymnastics.