Slightly Spoiler-ish mostly I'm referring to chapters that are already up here, but even the discussion itself implies some stuff about the new chapter. I've blacked out specifics tho.
As far as I can tell the new chapter was bonus content in a limited edition of volume 7, and also being sold separately (at least electronically), but not part of the standard vol 7 or serialized in Yuri Hime. If it is intended as a forced apology/appeasement to the bulk fandom it is very poorly executed in terms of reaching them.
It also seems quite carefully (lovingly?) written, and consistent with the main story (characters and themes). I'm more inclined to think it was a self-motivated effort by tMnR expressing the ending she wanted for the characters, a bit outside the bounds set for the main story (a bit like Tachi has done extra stories on Yuu and Haruka after Sakura Trick).
I've said before that I consider the main story was about Uta accepting her feelings for Kaoru as valid and legitimate, that she wasn't betraying anyone by having them (not about having those feelings reciprocated, or even pursuing that reciprocation); it makes it quite a queer story, since heterosexuals rarely end up questioning their feelings and identity on that level. That story was resolved in ch36 with Uta embracing her feelings and also getting acceptance from Kaoru who even went so far as to equate them to her own selfishness about the nature of their relationship.
Coming after that resolution it is definitely fair to consider ch37 as an epilogue, albeit one with a weirdly ill-defined ending, presumably in an attempt to keep the romantic/platonic question open. Although even then it still felt more like "veneer of deniability over romantic" than "possibility over platonic" (eg making a point of Uta re-affirming her feelings in the middle, and the waiting wife vibe from Kaoru at the end), so I suspect romantic couple was tMnR's personal preference all along.
I think the "vision" for the story was about accepting their feelings and dealing with their issues, and in that frame the nature of their final relationship is not important to the story as a whole. If that was the case one can see a couple ending could distract from the main point of the story, and why the author might want to leave the ending ambiguous.
Slight spoiler However with the extra story showing up so soon, making it clear a couple ending was something that tMnR was ready and willing to write, I have to wonder if it was actually Yuri Hime who didn't want it, which would explain a lot about the roughness at the end of ch37 - putting in as much implication as possible with out actually confirming them as a couple (like Korra and Asami in Legend of Korra). It was a bit weird as an ending, but it was a good set-up for the extra chapter.
It seems strange, but editorial influence seems like one of the more plausible explanations for the end... I think I've asked this before, but just how big of a taboo would there be in Japan about "stealing" your brother's wife this way? In the West people might shake their heads a bit, but younger people wouldn't get too upset about it especially in fiction. Yuri Hime must have seen the end well ahead of time and surely gave feedback. If they wanted a couple ending, surely they would have asked for one, so it seems like the implication is that they didn't want it (unless manga publishers are completely "hands-off" which also seems pretty unlikely given their financial interest in keeping readers happy).
Somewhat bigger spoiler nice to see Risako again; I was a bit hampered by my limited Japanese, but it seemed like she was quite cheerful teasing Uta about her couple-y dynamic with Kaoru
Massive spoiler And the cute way that the initially mature and composed Uta completely overloaded after Kaoru kissed her: 夢じゃない なにこれ 夢じゃない!!! (not a dream; what is this; not a dream!! while pinching herself) lol
As others have said, it is lovely, anyone who refuses to read the chapter is only hurting themselves. ChaosTeam have announced that they are working on it, so we should be seeing an English scanlation soon!
