@Willie the Duck - awesome, in that case I would also recommend Savage Worlds. Here’s why:
1. Focus on character personality, roleplay, perhaps more robust general non-combat/non-spell resolution mechanics. Certainly better supporting* adventures where treasure hunting and combat are, at most, a subset of standard activities.
The latest version, Savage Worlds Adventure Edition (commonly referenced as SWADE), is really quite good in this regard. Characters can pick Hinderance (which give them ‘build points’ to buy other things at character creation) which could be physical, mental, or social complications. These help put some mechanical weight behind characterisation, without being as strongly focussed on these as a game like Fate would be.
The game also has a simple system for Networking - i.e. talking to people to get information, and this can be stepped up into a Dramatic Task if you want more emphasis on it. Dramatic Tasks are great in general. They give you a system where players can use their skills and creativity to address non-combat challenges with a level of detail comparable to combat (not quite as crunchy, but close), meaning these types of encounters can have a similar, satisfying, game component rather than being completely free-form or just boiling down to a skill check. One of the really nice things about Dramatic Tasks is how they mesh with all the Edges (think a cross between feats and class abilities in D&D terms) that effect initiative and so on. Dramatic Tasks are IMO the best non-combat resolution system I have come across in 40 years of gaming.
Finally, there are Interludes, which are intended to represent those quiet time where characters relax and talk around the camp fire etc, and the GM nominates a player to tell a short back-story about their character in return for a small meta reward.
2. I think Fantasy, Sci Fi, and urban fantasy would all be natural fits. Maybe Horror and Heist as well. Probably not anything like specifically anthro, giant robots, wacky comedy, or non-fantastical historical dramas or such.
SWADE has all these covered out of the core book alone - it is truly one of the most complete core books of any game I have played. There are Companion books which add more detail for specific genres. Fantasy, Supers and Horror are already out and Sci-Fi is probably due next year. SWADE also has one of the best third-party support communities of any game, definitely top five, maybe even top three. So there is a crime focussed third-party campaign book, for example. And many third-party game worlds with new edges, equipment, adversaries and so on; all to a good standard in my experience.
3. As much as I would love a 1-10 page ruleset, I think the general consensus is a preference for not-rules-light. At the same time, I don't think anything like GURPS, Hero System, Shadorun 1e, or Aftermath is necessary. People are used to 3e/4e/5e level of mechanical-character-differentiation, so a reasonable number of toggles to play with.
I would put SWADE as very comparable to 5e in terms of mechanical widgets to play with. The advantage is that a lot of these things are opt-in so you can start with just the more core rules and build out from there as you go. For example there are rules for called shots, disarming foes, grappling and so on but you don’t need to bother with those until the moment comes. Such manoeuvres are very rarely locked behind Edges so anyone can try them when they need to; this avoids build-traps and lock-outs.
4. I think a sense of adventure, where leaping into action is as often as not the right response, is a plus. Combat shouldn't be excluded, I think.
SWADE’s moto is ‘fast, furious, fun‘ and is built to simulate pulp adventure of every kind by default. Pulp fantasy, pulp sci-fi, pulp horror and so on. Some people will argue they don’t find the game fits its brief but in my fairly extensive experience it does. You can adjust the dials and make it more gritty or more heroic but pulp action is the default. Another interesting feature of the system is that characters start out fairly competent and resilient and while they do grow through progression it isn’t as marked as in D&D so play remains manageable throughout. It’s like they took 3-12th level and dropped the rest (though again, in some of the expansion things can get much more gonzo - the Wish spell now exists for SWADE, for example, and there is an excellent official adaption of Rifts using SWADE, too).
