Maybe FATE would be worth checking out?

Things I think it might hit for you:

• Characters are good at what they're good at. You define the core concepts of your character and use them to get bonuses. Your character is a Highly Trained Ninja? Then yeah, you'll be getting bonuses to all your sneaking, hiding, acrobatics, flashy martial arts, etc. Plus, the way the maths work, the dice have a bell curve centred at +0 (extremes of +4/-4) so the +4 in your character's best skill is constantly having a huge impact on outcomes. Contrast that with your +7 in D&D which is still missing 25% of the time

• Fairly simple rules. The core is, describe what you're trying to do, and then use one of four basic actions to model it if a roll seems appropriate: Attack, Defend, Overcome (beat a DC) or Create an Advantage (alter circumstances/environment/characters to tip the odds in your favour). However, there's a little more for combat and also a meta currency to manage, which I'll talk about below

• Very quick to get off the ground. Character creation can take only minutes if you want. No mucking about with long lists of feats and spells and class builds and whatnot. You're actively encouraged to leave spots blank and fill them in during play when an idea strikes you, great for new and unsure players

• Completely setting agnostic, it's flexible enough to do almost anything

Things I think it might miss for you:

• FATE's approach is much closer to a story game, especially compared to something like D&D which leans towards the simulation side of the spectrum. Its meta currency, Fate points, aims to emulate the feel of an action movie or TV show. Spend points to do awesome things, get them back for accepting challenges, complications, and setbacks in your character's arc. That latter point especially often means the table needs to have a "writer's room" mentality, which isn't a good fit for all players.

• FATE doesn't really try to do certain things that D&D does, like strict resource management, accumulation of powerful loot, big powerful character level ups, or dungeon crawling. It can be done, and guides are out there to help you do so, but you will be bolting a lot of extras onto the system, so watch out if those are what you enjoy

• Which brings me to the last point, FATE is a system that really wants you to hack it and make it your own. It's very resilient to this sort of thing compared to something like D&D where getting some maths wrong can make things unfun in innumerable ways, but it does take effort and thought regardless, which may not be to everyone's taste. For example, you won't really find a "bestiary" of monsters to throw at players, you'll be making them up yourself, maybe entirely on the fly.

The rules are all freely available online or in pay-what-you-want PDFs. There are three current editions:

• FATE Core, all the rules of the game plus lots of extras, examples, optional systems, things like that

• FATE Condensed, all the rules same as Core, but with most of the extras cut out and overall streamlined down to 60 pages from 300 or so

• FATE Accelerated, uses the same basic ideas for its rules, but simplifies things down to the barest of minimums, e.g. dropping the skill list for 6 basic "approaches", simplifying the damage system.

Here's a link to FATE Condensed, as I personally found it easiest to start with: https://fate-srd.com/fate-condensed

One disclaimer: I haven't actually played it myself yet, but I have been prepping a one shot I'll be taking my D&D 5e group through this weekend to see if it's going to be a good fit for us, so I have been doing a lot of research!