There are a few conflicting mechanisms at work in classic Dungeons & Dragons which have been ported over to OSR wholesale.
High level characters need to be powerful and capable, they are after all the reward for a campaign well played. For high level characters to feel powerful, the low level characters need to feel weak, which can be a problem because these games are supposed to be open and permissive: if the players come up with a good plan you should let them try it. But if their plan requires that the group’s thief climb a wall (25%), move silently (15%) and backstab an ork, killing him before he can raise the alarm (~5%), how are you supposed to let them pass?
If you make the low level characters capable enough to do these things what effect does that have on the high level characters? This zero-to-hero progression is one of the defining features of the D&D family and there are very few other games which have managed, or even tried to make it work. I’ve lost count of the number of RPGs I’ve played in which just broke after ten or twenty sessions because their PCs were too powerful.
The other, related, issue with D&D and OSR is niche protection. You want your fighter to be the best at hitting things and your thief to be the best at sneaking, but you also want the thief to be able to hold his own in combat and the fighter to be able to sneak when he needs to. But if the thief can only move silently 15% of the time, what does that leave for the fighter? 10%? 5%?
It’s frustrating seeing all around the internet fixes and hacks discussed to sort out skills, or add them in if they were entirely absent in the discussed edition. Frustrating because often they are fixes for one class, screwing over the others, or they are fixes that benefit low level characters which get unwieldy at high level.
Most of these problems were ‘solved’ in the ’80s with systems which did away with classes and levels (Runequest or Star Wars for example). This helped make sure every character was individual and stayed relevant, but it isn’t an option in OSR as without class you don’t get that immediate entry point when your class provides you with a package of powers and responsibilities, and without levels you can’t have a character who will start off scared of a goblin with a stick and end up coolly taking down a dragon and it’s army of undead.
So what’s the solution?
Stop adding new rules to the mix for one thing. If the rules you’ve got aren’t working either make them work or replace them. This is where the problems all started, back in Greyhawk when they added the Thief and his skills to oD&D.
I had a hack myself a while ago for Basic D&D which with one simple tweak fixes non-thieves trying thief skills, it fixes low-level thieves being a bit rubbish and it wouldn’t mess up high level thieves. It even works best with the annoyingly slow 36 level thief progression in BECMI and the Rules Cyclopedia. I call it Everyone’s A Thief, Including The Thief or EAT ITT. This is it:
All characters gain Thief Skills as if they were a Thief of their current Level. When a Thief rolls on these Skills though, they roll with Advantage.
That’s it, the whole thing. With those two sentences all Players have a clear idea of what their character is capable of, thieves are more useful and because of how advantage works it doesn’t overpower high level thieves.
I think I can do better than that though. Class&Level *are* D&D. There is nothing more important for your character than those two details. So lets use them:
When your Character attempts a task, roll d20+Level, if it is related to your Class roll with Advantage, if you roll higher than the Defence of the target or the Difficulty of the task your Character succeeds.
My mantra is “Make your rules do more,” there’s 90% of D&D in one sentence.
If you want, apply Ability Mods to rolls as appropriate. Saves would just be a Difficulty 15 roll but you could give bonuses to Class or Race if you want. Add in XP progression (lets just give them all the Fighter’s) and Hit Dice and your favourite magic system, and now it’s playable.
Anyway, I’m not trying to write another retroclone here, I’m just trying to make the point that with a bit more thought and subtlety we can fix D&D issues without losing D&D strengths. Otherwise we’ll just retread all of the Eighties and before we know it we’ll all be playing Vampires and Werewolves in the World of Dimness again, and nobody wants that.
Ominosity 
From a maths perspective, things like “roll d20+Level, or with advantage if it’s related to your class” sound great, but they need to be examined.
How much more powerful does this make the difference in level (and how much difference do you want it to make)? Do you want the gap to widen as levels increase? Do you want class to dwarf stats, or should stats still be relevant at high levels? Do you want a nice simple formula (that might give poor results), or are you willing to have a table that says what a bonus people should have is? Also advantage is worth a floating +1 to +5 (average about +3), but it also reduces variance, so people with advantage can more reliably predict the outcome. Is that what you want, or would a flat +5 maybe work better?
4e D&D is the best example of figuring this out, everyone progressed at +1/2 level, with a +5 for being skilled in something, and another +2 (I think) for things you really focussed on. Then 5e did the same thing but took out the +1/2 level, meaning low level people could still largely compete with high level people, but training mattered. Ignoring the rest of the design, the actual numbers they came up with worked pretty well, and could be ported across into OSR games without subtracting from what makes them interesting.
I am aware of the effects of what I’m suggesting. As penance for doubting me I require you to produce three A4 sides of charts and infographics detailing everything you said above.
I chose d20+level so that every level of increase matters. Sure, if you have characters gaining levels at the same rate then when combined with the effects of the Advantage fighters will grow in ability significantly faster. Two things to consider though; thanks to the way advantage works this effect will be most pronounced when the fighter is out matched and then if you slow down level gain slightly then fighters will progress more in ability than HP bloat. Both of these sound like fun tweaks but they are only options and easily tweaked in another direction if you like.