Lots to add here.
First off, there's active system support and passive system support. Not all support need be active in order to be present and effective.
But "passive support" is not the same as "doesn't prevent." It still needs to be, y'know,
supporting something.
Rules for mundane and magic item pricing, for example, passively support an economics- or industry-focused game. They don't actively support developing trade between two places (that would require price differentials and a bunch of other stuff), nor do they actively support the
creation of magic items. But they are at least a starting point for doing those things yourself. Your tools and materials to work with, metaphorically speaking.
"Simply getting out of the way and doing nothing to discourage it" is not and cannot be support, of any kind. It isn't opposition, to be sure, but it isn't support either. It is not passive support, it's very intentional nothing. Because, by that standard, literally every game ever made "supports" roleplaying, because there aren't any rules that
prevent roleplay.
Doing nothing to prevent something is not supporting it. It is
permitting it. Permitting is a far cry from supporting in most cases. Chess
permits roleplay, but that doesn't mean it supports roleplay, passively or otherwise.
Does D&D support slapstick play? Yes, passively, by simply getting out of the way and doing nothing to ban or discourage it. Same for some definitions of character-driven play.
To counter this with what I would consider actual "passive support": fumble tables, potion mixology rules, magic item quirks (e.g. the rules from 13th Age), and actually useful personality-trait systems would be
passive support of "slapstick" play. That is, none of them is in any way doing something intended for slapstick use, and none of them is
directly applicable as slapstick itself. But they provide raw materials which could be taken up, adapted, rearranged, and/or extrapolated in order to produce a "slapstick" tonal experience.
Second off, the OP lists 8 or so play styles and yet still misses several important ones, which may overlap with both each other and those on the original list:
I think part of the problem here is that we're conflating at least two very different meanings of "style" here. One is about the overall gameplay intent or concept, e.g. "problem-solving." The other is about the manner/method. A possible third is about the tone, since "big damn heroes" and "mercenary" are both tonal things, not relating to purpose/concept ("problem solving," "hack-and-slash," "political"), while "sandbox/freeform" (though, since "freeform" is rather loaded already, I'd prefer "impromptu") and "adventure path/railroad" are both about the manner or method by which the campaign is played.
That is, I don't think it is in any way a "hybrid" to speak of a game as a "big damn heroes" "political" "sandbox/impromptu" game. Those three descriptions seem to be operating on distinct levels. Likewise, a "slapstick" "mercenary" "adventure path/railroad" game sounds perfectly plausible.
In general, D&D supports degrees of heroic or mercenary tone for various reasons, and doesn't handle other tonal types as well, e.g.
Zeitgeist had a lot of heavy lifting to do to make it make sense that the players are formally part of an official law enforcement agency. Such a tone is not actively supported, and I would not personally call it even passively supported, but it is certainly permitted. Likewise, while it is
possible to do horror with D&D, it's rarely even passively supported without significant bolt-on modules added (see: Ravenloft), made more or less from whole cloth.
For intent/concept, one that hasn't strictly been mentioned thus far:
Survival. This is an important one, because 5e (much like 3e and PF1e before it) is in fact actually set against this gameplay concept. You have to actually go in and
delete existing rules, and often add more rules as well, to get a survival-focused game. It's a complaint I've seen quite frequently from old-school fans. Conversely, despite the many claims to the contrary, 4e actually did passively support "survival" gameplay, via the Healing Surge and exhaustion mechanics, and Dark Sun then built on those rules to bolt-on actual active support for it. I lack sufficient experience to speak of previous editions, but given the strong association between gritty survival-based gameplay and the OSR, I can only imagine that the rules were
at least passively supporting it back then.
This way, we can differentiate active support (rules specifically designed to achieve the described end), passive support (rules
not designed to do that, but which can be re-purposed for it, or used as an effective foundation), mere permitting (more or less lacking any rules at all relevant to the topic), and outright opposing (having rules that actually resist or prevent something.)
Of these, D&D actively supports the second and third, passively supports the first and fourth, and gently fights against the fifth unless the DM does some tweaking.
See, here I actually agree that it passively supports "mercenary." It's not just that it doesn't do things which prevent this; it does, in fact, lay ground work for doing so, but little more than that. See, for example, the oft-mentioned lack of stuff to
do with the filthy lucre obtained from your mercenary ways. Conversely, it simply
permits sandbox play; there's really not much in it which can be purposed toward its specific ends. For example, if there were robust support for random generation of things like settlements, businesses, organizations, and NPCs, then those things would be good passive support--they make it easier to develop locales on the fly, which is something sandbox play wants to do a lot. Active support would look more like rules for how to design maps (with specific reference to hexes vs grids), ways to spice up an existing locale, checklists for making sure you're covering your bases (literally and figuratively), and wandering monster tables of varying levels (preferably with advice for how to build your own such tables.) As it is, the DMG is....pretty sparse on the NPC-generation front, and nearly silent on the rest--that's pretty squarely in the "permit" category, with just the faintest blush of passive support.