I think you can definitely move 5e more towards a group based adventure oriented sandbox. I have several games I would put above it on that list, but with a decent amount of effort it can work.
When it comes to more protagonist oriented play it has things actively going against it. First of all characters in most iterations of D&D feel like space aliens to me. They have barest of connections to their environment. Backgrounds are a plus here, but are firmly in the past. Characters are unmoored. Secondly characters are entirely too specialized in 5e. Outside the confines of a group characters are not very capable of making their way through their environments.
That's not really criticism from my perspective though. The game is damned good at what it does. So much that it's fairly resistant outside of that area of strength,
Yeah, the main thing with trying to shift the game toward more player directed play is to somehow come up with existing connections and goals that give the PCs a sense of place in the world. The Background does this a tiny bit, and the Bond can potentially add a bit more, but these are like bare minimums.
For me, our 5E game started off with the adventure from the Starter Set, and when everyone said they liked the game, we continued, so that became a big part of things. The town of Phandalin and the people there became foundational to everything that followed. They started a trading company and sought to expand, and that's been a main theme all throughout.
We also folded in a lot of existing lore from our previous campaigns, going back to our earliest days as a group together, which was at the time that 2E first appeared. So there was blatant nostalgia going on in that sense, but it hooked them and motivated the players and their characters in interesting ways.
There is still a central threat, or Big Bad, that they're working against, so that's a pretty classic GM driven element, but when and how they engage with that is largely up to them.
A lot of this is loose. I don't have formalized mechanics. Honestly, most of it is just a longstanding group whose members are familiar with each other and who trust each other playing the game the way they want. It's a case of the social contract overriding the game and allowing things to be this way. This is what blinded me to a lot of the flaws of 5E and I expect is why many people think that the game "supports" more than one style of play.
I don't think most people would think that an example of "supporting a playstyle" would be to "actively ignore almost all the advice in the books and many of the rules and expectations".
I actually have the opposite opinion of PF2. It's terrible at adventure paths. It's pretty strong at providing a skilled play environment and it's more broadly capable characters can excel in games that drift more towards active protagonism. It's not good for GM plots in my opinion even if it's what they have been trying to sell.
I grew to dislike Pathfinder 1 so much that I have no desire to try Pathfinder 2. I'm surprised to hear that it leans more toward skilled play though, and away from Adventure Path style play. Why do you think this?
What, how, why? What would you need to moor them?
Because people tend to be moored? Most folks have connections and relationships and goals and so on. Having game elements that emulate that can help. Yes, this can be done without rules.....but it can also not be done. As I said above the "Bond" bit from 5E is pretty weak. Other games require a far stronger connection to be offered, and usually give it more weight through mechanics connected to it. Certainly more than "if you play your character like the Bond you've chosen matters, you get an inspiration die".
When a game specifically tells you to do something, it's usually an indication that thing is an important part of the experience. The same way that XP is a strong indicator of what a game is actually about. These things being mere suggestions rather than specifically stated requirements shows that there are not essential.