First, I haven't seen anyone say this as bluntly as I think it needs to be said. Games that don't have skills, and even some that do, should be run with one major principle in place: characters succeed. If it's in question, roll a d6 (or maybe a Saving Throw in Swords & Wizardry and related games). If it involves people's reactions, make a reaction check.
When a character in D&D in any of its varieties before 1.5E tries to do something, it should generally just work unless it's ridiculous. This is even more true in versions that don't have a variation of the Thief class. No Thieves and you want to pick a lock? Did you buy lockpicks? Then you do it. Maybe a roll of 5-6 on 1d6 will cause the lockpicks to break after opening the door. Want to bust down a door? You do it (with a wandering monster check). If the DM says that the door is particularly tough, roll 1d6 with a modifier for your Strength, and a 5 or higher busts it down. If it's particularly complicated (hunting through a library for a particular manuscript, for example), maybe refer to the character's background and make a ruling based on that. In fact, setting a general background (Conan the Librarian!) might be the one thing you could add to character creation, where AD&D 1E has Secondary Skills for much the same purpose. Something technical like forging a sword would require an appropriate background, for example.
The same principle applies to other games. You're playing Traveller and you want to rappel down a wall? Do you have some rope? Then you do it. You want to convince someone to help you? Roll a reaction check (if you have some appropriate skill, maybe that will help modify the roll).
The general idea is that characters are generally competent to carry out the plans that their players make, for the most part.
Anyway, for the second point, here's a list of games that I have come to think have really stood the test of time, as it were. Games that I still like on the whole. A lot of these games work a lot better if you apply the above idea:
Amber Diceless Roleplaying
Call of Cthulhu (up through 5.5E or so)
Chivalry & Sorcery (1E and 2E, though there have been some recent unofficial versions based on those that are great too)
Cyberpunk (2013 and 2020)
D&D/AD&D (up through 2.0E/Rules Cyclopedia; after that, it gets weird; retroclones and similar variants elsewhere)
Empire of the Petal Throne
GURPS (3E revised and 4E)
James Bond 007
Lace & Steel (I think; I only played it once, still it looks good)
Lords of Creation
Pendragon (I prefer 4E, unlike other people apparently, but the other editions are great too)
Rolemaster/Spacemaster (I guess they're calling the good edition "Classic" now)
RuneQuest (never met a version I didn't like, yet)
Space 1889 (the GDW version, and I actually like the system; it's pretty much the only time other than WEG Star Wars that I don't mind dice pools)
Star Trek RPG (FASA; I can't say that it's great, but it does the job well)
Star Wars (West End Games; one of a very few dice pool games that don't suck)
Star Wars (WotC, though it isn't as good as the WEG one; nearly the only use I have for D20 system)
Stormbringer (4E is the best, after that they messed it up)
Top Secret (original, the 1981 second edition, and the Companion)
Traveller (classic LBB77/LBB81/TTB/ST and MegaTraveller)
2300AD (I actually like the system)
Unknown Armies (I can't run it, apparently, but the ideas are great; maybe I'd be better at running it today)
Villains & Vigilantes (I haven't played Mighty Protectors yet, so I can't speak to that edition)
Games that I haven't played, but look good:
Big Eyes, Small Mouth (specifically Sailor Moon, Ghost Dog, and Revolutionary Girl Utena)
Dungeon Crawl Classics
Stars Without Number (I'm pretty excited by the Revised Edition)
Starships & Spacemen (2E looks better than 1E)
Swords & Wizardry (White Box mainly, but the others are OK)
White Star (S&W: White Box in a galaxy far away that isn't Star Wars we promise)
Realms of the Unknown
You'll notice a distinct lack of White Wolf games, Shadowrun, and Burning Wheel. That's because dice pools are dumb, in general. No Hero System because it's just too scattered for me these days. Also no Savage Worlds, which I might be being unfair about - I suppose that I have to play it more. And no goddamned FATE.
I might have missed some. If I didn't include your favorite system, ask me why I didn't. It might be an oversight.
NOTE: As time has gone on and I remember more games to include, I have been adjusting these lists slightly. I've also removed one because the author chose to associate themself, even if outside of gaming, with alt-right personalities. The alt-right may be "just joking" about their white supremacy, but I don't find it funny. I've removed another pending the reaction of the company to abuse allegations in regard to a particular individual.