I've played a fair amount of D&D, but I've always enjoyed other RPGs quite a bit more. There's a huge variety of games out there covering every genre and playstyles ranging from the densest exercises in tactical spreadsheet juggling to storytelling fare so light it's more like improv than gaming. My tastes lean much more towards the latter. Try looking up the FATE Core system, there's a lot less dice and rules but just as much fun and adventure.
One of my biggest complaints about the games I play is the level of dice rolling for even tasks that your character should be perfectly competent in. To me, it breaks the immersion of, say, heroic-level skills in sneakiness when some random untrained person always has a chance of noticing you. I think some of the later versions of D&D "fixed" this with mechanisms like taking 10 and such, but as long as 20 is always an automatic success, that's a 5% chance of failure all the time.
Is this something that FATE fixes? If so, I'll have to look into it!
5e addresses this by assigning much more responsibility to the DM. According to the rules as written, the DM should only call for a skill check when there's _possibility_ and _consequences_ for success/failure.
On top of that, the way they rebalanced skill checks and AC ("bounded accuracy") makes the game feel a lot better. This [1] is a fantastic article specifically addressing those changes, if you're interested.
That's a rule annoyance for me as well that has been mostly fixed by 5e, and doesn't exist in most other systems I've played.
Examples:
FantasyCraft: D20 based, but drops the automatic success/failure rules, so at heroic levels you can have guaranteed success.
Shadowrun 3E: The maximum skill rank a starting character can have in any skill is 6, which translates to over 99.99% chance of success on "easy" difficulty checks and over 98% chance of success on "average" difficulty checks.
Earthdawn: Every 5 or 6 steps of a skill adds one more difficulty level that you can succeed automatically (starting at step 3, where you auto succeed at difficulty 1)
It's worth noting that technically 20 is only an auto-success on attacks and saves. The way you describe is a very common house rule, but it seems to frequently be an accidental house rule (more of a misreading).
Has it changed over time? It looks like what you say is true in 5E, though only for death savings throws. But I played 3.5E. My memory isn't good enough to know whether that makes a difference.
I looked it up in the various books I have on my shelf:
D&D Redbox:
No skill checks, Attacks succeed if the modified value is 20 or higher. So a level 1 Str 9 character can never hit AC -1. A level 1 Str 18 character needs only a 17 to hit any AC. These rules are different in later rulebooks when the THAC0 is introduced.
[edit] Checked my Expert rulebook, and found that the modified 20 always hits is just due to very low ACs not being represented in the Basic charts (In OD&D lower AC is harder to hit). The target number to hit goes up as AC goes down, but "sticks" at 20 for 5 points of AC. There is no special rule for auto hitting or missing with a natural 20 in OD&D. Note that monsters are quite good at hitting with the weakest monsters hitting about as well as a mid-level fighter with mundane equipment.
D&D 3E: Only attacks succeed/fail on a natural 20/1.
D&D 3.5E and Pathfinder: Attacks and saves succeed on natural 20/1
Too late to edit, but thiefs do have skills, and these are d% and always fail on a 100, which is a 1% failure no matter how good. Also, I forgot how bad pickpocketing is; if the opponent has HD equal to your level you have less than a 20% chance of success throughout the intermediate levels.
It is something that FATE fixes, albeit by handwaving for the most part! FATE is a much more story-focused rather than mechanics focuses game, and one of the core concepts is to only play through dramatically interesting conflicts with actual consequences for success and failure. In such a case, a routine skill check in D&D would just be explained as flavor in FATE; if you are breaking into a common shop as a master thief, just describe how you do it; now when you come across the merchant and the necromancer conspiring in the back, then the dice come out!