So some thoughts based on recent posts......
I think when it comes to theme, the genre and setting will play a big part, for sure, but even still, there's usually leeway to explore other themes than just those imposed by setting. And some themes can be pretty universal, so you can take them and drop them into any genre or setting.
I think what can be tricky, whatever game we might be talking about, is that there's going to be a call to action of some sort, and a general need to work as a group. These things can be challenges to examining theme through an individual character.
Take 5E D&D for instance. The published material makes it pretty obvious that the expectation is that PCs be at the very least inclined to heroics; there's little motivation in each of the published adventures to engage the characters beyond the call to heroism, and perhaps the call to adventure. It's simply expected that the characters band together and then proceed to try and save the day. These can be added or altered by a specific group so that it's less altruistic or so that the goals of the scenario as published can somehow coincide with personal goals of the PCs. But it's pretty constraining to kind of have that expectation and motivation be inherent.
The fact is that the group element is already a strong constraint on what you can do with a character thematically. And this is one that most games struggle with, to some extent; RPGs are (mostly) a group activity, after all. Adding more restrictions like "only good alignments" and so on just makes it even tougher.
My 5E campaign has elements of this, for sure. I try to put it at odds with other goals, though, so that there's some tension. Some decisions that need to be made that aren't always easy. And each PC has goals of their own, as well, so I try to bring those to bear. The game doesn't do much in this regard, so it all boils down to the players giving me ideas and me using them in play. We know each other well, and I have an idea of what they'd like to see, how they want their characters to be challenged, and so it works well enough.
But it's far from the optimal game for this kind of play, and even with as much as my players and I try to put this stuff in there, the thrust of play is primarily heroics. In an earlier edition, the thrust of play might be in raiding a dungeon. In this edition, it's about stopping the bad guys.
If you're examining a theme of some kind with a character, then I think there needs to be risk involved. I think this came up a hundred or so pages ago, but I think it's really relevant. The risk of failure needs to be real.....if your theme is one of redemption, then that's something that the character must be able to either succeed or fail at. It shouldn't be as simple as a player deciding at character generation "My character is trying to atone for his horrible past" and then simply plays the character as they would just about any other, engaging in the group's goals, and occasionally monologuing about their sordid past. If there is no struggle for the....if it's all simply up to the player, then it's not much of an examination of redemption.
Now, this doesn't require mechanics; it can be done simply through play, with the player deciding things for the character. But in the absence of mechanics that can help here, then things need to be framed by the GM so that there are consequences that matter at stake. There needs to be some kind of drawback, whichever way the player decides to go. There needs to be a challenge.
I think it's only through that kind of challenge where we get to actually examine a theme instead of simply using it as some kind of character trait to help us roleplay.
Anyone can play a character with a haunted past and use it in their portrayal of the character. But if you want to actually examine trauma and its effects on someone through play, if you want to see if a character can overcome their past and gain some sense of peace or redemption, then you need to go beyond just playing the role. That theme needs to be a focus of play (shared as play may need to be among the group).
To me, that speaks of, if not more agency, then maybe a deeper agency as a player.