I've been meaning to do something like this for a while, but. Welcome to my big "Su-won's sexuality" post! I'm going to gather all the textual evidence (that I can remember) in one place and give my personal thoughts, which. Hopefully people will find helpful.
This is one of the first ones! Kye-sook specifically says that although he's receiving marriage proposals/held discussions, Su-won has simply never shown interest in "such things" (i.e., marriage and romance/sex). Once we've gone through the canon examples I'll give my overall thoughts but keep this in mind for now.
The rest of my post is under the cut!
Su-won veeeery specifically has to correct Geun-tae and Mundok here that he and Lili are not a couple, have not been having sex, and are not interested in one another. In the previous pages, the assumption that they're a couple actually catches him off-guard, because presumably, he had no idea where the conclusion came from.
And then here! When Lili asks if he has anyone he likes, Su-won specifically responds that he doesn't understand love and romance. Note that he specifically also adds that he likes people still, and Lili remarks that she felt Su-won was being honest here, which I would agree with.
And this! Su-won does the classic "yes I like this person! I also like--" to make it clear he's not talking about romantic feelings here. Hak's assessment of Su-won is that he holds equal love for all things, but while I get where Hak got that, I do think the narrative disproves him in that Su-won does care for particular people more than others, just not in a romantic sense. I do think he's picking up on Yu-hon's influence, though, which, I will derail this post a bit--
Specifically, Yu-hon teaches Su-won not to think of others as people, to discard them readily if they're unneeded, and even cements it with a "the kingdom will suffer if you don't." Su-won internalized the hell out of this, so even if he does care about particular people, he won't prioritize them because of the teachings he's taken from Yu-hon (as we see in chapter 1 where he betrays Yona and Hak, despite them clearly being dear to him).
And I can also add a bunch of support that Su-won does indeed care for Hak and Yona in particular, but I think canon emphasizes his platonic bond with the two of them throughout it enough that I don't need to lol, so I'll get back on track. The main point is that his tendency not to prioritize or seem to care for particular people is something that was taught to him, and it's separate from his lack of romantic interest in people--we can see he still develops attachments to people in spite of Yu-hon's influence, so if he was inclined towards romance, he'd still develop romantic feelings with or without Yu-hon's teachings.
Lili again remarks on the lack of interest that Su-won has in women and romance, to the point where many people have suspected he's gay (lol). He also makes the interesting comment that "any woman would do" to get people off his back--though in the official translation it's more ambiguous and reads more like it's the thoughts of those around him and not his own--and Su-won also remarks that he's happy to have Lili around, while also not being interested in marrying her due to the lack of political gain, because it gets people off his back.
And then the final "Su-won just doesn't feel romantic emotions" which is a pretty telling piece of dialog lol.
So, what does all this mean for Su-won's sexuality? Well, I think we have enough support to conclude that Su-won is a-spec, mostly likely aromantic or aro-spec. I think since Kusanagi rarely expands on characters' sexual preferences and it's common for the romantic to imply the sexual (in manga/literature, not necessarily in reality), you can also pretty easily conclude he's asexual or ace-spec if you want, too. (Edit: He's also shown to avoid sex as much as romance in canon, so although there's not a definitive statement about his feelings on sex due to that just being a thing the manga doesn't do, I do think ace-spec is a part of his canon identity, as well. Though again, I think a lot of different a-spec identities fit him!)
The text doesn't explicitly comment on whether or not he's attracted to men, but I don't think him being gay is the actual intended reading to all of this, either (see "he doesn't feel romantic emotions" instead of "he doesn't feel romantic emotions for women"). We have a character that expresses an aversion to marriage to women but not an aversion to romance in general in Kija, and his handling and Su-won's handling feel very different, given Kija still maintains an interest in romance and has all sorts of homoerotic subtext with Jaeha, so I feel like we would've gotten some of the same subtext if Su-won was intended to be more interested in men than women. To me, it seems to be more of a case that he's equally uninterested in all genders, and the fact it was brought up "any woman would do" and "I'm only weighing the political benefits of marriage" indicates he only views his own marriage as a political tool and has no actual romantic interest in one woman over another (or men, for that matter). Lili even comments that Su-won is entirely uninterested in women, too. All this just lends itself to reading him as a-spec to me.
Note that he does still care a lot about people, but it's his platonic bonds with people like Yona, Hak, Lili, Kye-sook, and Joo-doh that are always front and center with him. I also think that although the intended reading is a-spec, you can make an argument for a lot of different identities that fall under that umbrella! Canon is vague on the exact specifics of his general disinterest in romance/sex and identity, so that means there's a lot of room to play around with different headcanons and specifics.
Noooow I do think this fandom doesn't know a whole lot about asexuality/aromanticism so let me add that it's a spectrum. You can have demiromantic, greyaro, and aro-spec people that still feel some romantic attraction in the right circumstances, and same with demisexual, greyace, and ace-spec people who might feel some sexual attraction in the right circumstances. These are separate spectrums, too, so you can have an aroallo person who still feels sexual attraction or an alloace person who still feels romantic attraction. (Allo just means feeling attraction outside of being a-spec.)
And that doesn't even account for aro/ace people who might also be sex/romance-favorable, which means even if there's no attraction, they still enjoy or want sex/romance. It goes from repulsed to indifferent to favorable! Also separate spectrums, so you can have an aroace person who is sex-favorable but romance-repulsed, for example. Queerplatonic relationships are an option too!
So yes! Even if Su-won is a-spec, you can still ship him! Many ways to do so in a way that's not 100% allo, in fact. Honestly I would love, love, love to see more content exploring him in an aro and/or ace light, so please try it out!!