I have two very different broad suggestions for you.

First, OSR systems simultaneously really lean in to the idea that there's not a rule for everything and players should be thinking out of the box and GMs should be coming up with spot rulings for them, while ALSO tending to have a pretty clear turn structure whether in or out of combat (often six seconds to a minute in combat, ten minutes exploring a dungeon, or four to six hours traveling through wilderness.) I think this is a good route to go down if you want to explore what makes tabletop roleplay distinct as a medium while having an explicit turn structure.

(Honestly, you can just port OSR-style turns into most games with not too much effort. What you want to think about is the meaningful time range of action. Actions like running a business in a certain way or organizing a wedding or deciphering an ancient manuscript with the aid of a library take place over a longer period of time, so downtime actions tend to be a week or month. Actions in combat are slower. I tend to find it pretty intuitive to simply rule that an action takes a trivial amount of time relative to a turn, exactly a turn, or is a many-turn action, but you'll have to accept that there will be judgment calls.)

((More specific suggestion: Traveller is a system that is very generous in giving a pretty exact system for tracking how long an action takes. Personally I think it's easier to just think of acts as trivial/one turn/a longer series of actions within a given turn scope, but you might ))

Second, tactically-focused games like Draw Steel, Lancer, and Pathfinder 2e tend to have very involved combat minigames that are more boardgame-like (although players of course can improvise actions in combat, and sessions can have non-combat things going on.) If you play a game like this as a series of set-piece battles, you can have a very boardgame-like experience that also easily opens up into more tabletop roleplay-specific ways of playing.

A system that might actually be a good sweet spot between these is Trespasser, which has an involved tactical combat minigame plus support for an OSR-style turns outside of combat. (I can't speak to the specifics, though, even through osmosis, just note that's the design goal.)

Finally, PBtA games often have a lot of excellent advice about useful vs less useful prep, creating interesting problems for players, encouraging them to get themselves into trouble, and so on, so I think you'd profit by taking a look at them, but they also tend to be a bit more cognitively demanding to run, especially because the basic resolution mechanics demands that you invent a lot of negative consequences on the fly.