I don’t agree that all skill checks are binary, many of them are not.
When my investigation guy rolls a 30 to check a room (and he does...a lot), you expect to find a bit more than if you get a 20, and more than if you get a 10.
Knowledge checks are often the same way. So that’s why I think 2d10 works for skill checks in a way that isn’t needed for attacks.
Actually, once the DC is set, it doesn't matter if you make it by hitting the DC or beating it by 20 points. However, people seem to think that a higher total is better and that "my 24 beats your 18" even though we both only had to make a DC 15 check.
One of the players in our group actually brought this up during our session on Sunday. He proposed something akin to the more you beat the DC or AC the better the result. The DM countered with "Sure, but the more you fail by the worse it will be for you as well. So, you roll a 1 on that poison save and you'll take maximum damage from the poison instead of average, but you roll a 20 and you won't take any. How's that sound?" It wasn't quite so appealing to him then...
That's why 5E stressed that a 20 won't always succeed on a save or skill check, even though many tables still play that way. Some things are simply beyond the characters and a lot of people don't like it so they play a 20 always makes the check or save, but of course then a 1 should always fail. 20 always hitting and 1 always missing is only true for combat, so that nothing is "unhittable" and no one "never misses".
And to DEFCON 1, for us we try to keep things lower key at lower levels. Typically your major "thing" will only be +2, maybe +3, so we do have a lot of +3's and +4's. Now that we are into Tier 2, we have a few +8's due to expertise, but otherwise +6 and +7 is more normal. We use a slightly altered ability score modifier system (9-12 = 0, 13-14 = +1, 15-16 = +2, etc.) so we'll have no +5's ever. Our DM tweaked the proficiency bonus a bit as well. You begin at +1 for first level, then +2 for 2-4, +3 for 5-7, and so on until +8 at 20th. We all agreed to these changes primarily because it seemed lopsided that a 1st-level character with a Str 20 (so +5) could have a +7 attack and damage, but a 20th-level character with Str 11 (+0) could only be +6 attack and no damage. With our changes, the maximum ability score modifier is +4 (at 19 and 20) and maximum prof bonus is +8 at 20th. The prof bonus is twice the best you could do with "raw" talent. It works for us and really doesn't affect the balance of the game any since your maximum adjustment is +12 instead of +11.