The Dashnaks are not the Armenian diaspora as a whole though. In OTL they were quite anti-Sovietic, to the point some its Berlin-based members supported the Nazis and participated in founding an ''Armenian Legion'' which fought in the Wehrmacht. Its unlikely the same thing happened ITTL, as I assume Turkey siding with the Axis kept them firmly on the Allied side, but I still doubt they are all too fond of the Soviets ITTL...More probably the Soviets. After the First Republic's collapse, the Armenian diaspora tended to support Soviets as a means to liberate Turkish Armenia (per my reading of Ter Minassian's book on the First Republic).
Moreover, I'd argue some other factors are pushing the local Armenians toward the WAllies:
I. At this point the collusion between Sivas and Moscow when it comes to the Straits is most likely an open secret, which is in and of itself a good reason for the Armenians of Constantinople to feel that they'd be better off under Greece than under whatever kind of international city the Kremlin has in mind.
II. During the interwar period the Greeks had plenty of time to work on polishing their relationship with the Armenian community of the city, to the point where the latter were arguably fully onboard with Enosis a few years ago. Granted, it was mentioned that things are a bit more complicated after these bonds were strained by three years of Axis occupation during which Athens couldn't do much in the city but it's unlikely two decades of work have gone totally in smoke. Moreover, the fact that the Dashnaks are stated to be the dominant political group in the Armenian community does show that the Soviets aren't in that great a position in that regard either...
III. In general it is fairly clear that the Armenians won't emerge as the dominant force in Constantinople so the best the Dashnaks can do is to try to sell their support for as much as they can, and I'd argue the Greeks, and the WAllies as a whole, are likely to be willing to pay more. The whole game plan for the Soviets is to try to leverage their very considerable overall power to resurrect the international city of Constantinople with them in as strong a position as possible. For that Dashnaks support would be good to have but it isn't really key: at the end of the day it is more about the Soviets' overall geopolitical muscles than about whatever the city's inhabitants want.
The Greeks, in comparison, probably need to go through the referendum shortly after liberation outlined in the deal between Stalin and the WAllies as well as find a way to win it if a Constantinopolitan Enosis is to ever happen. For that time is of the essence and they absolutely need the votes of the Armenian community. As a result, and as long as they don't ask for anything too over the top, the Dashnaks can pretty much give their price, both for concessions regarding the city's governance and in terms of Greek diplomatic position re Armenia, and Athens will pay it without blinking.
IV. Even for Armenians who have the geopolitical reasoning you describe this is probably the one instance where they want the Soviets to fail. If you want the Soviets to kick the Turks out of as much of Turkish/Wilsonian Armenia as possible having them pull off a move that is a) turning their focus for gains from their latest war with the Turks westward and b) have them build something of a working relationship with Sivas is the last thing you want to see. Even with Turkey having very few cards in its hands and the WAllies probably not too enthusiastic about limiting its territorial losses this will most likely end up meaning less territory being added to the Armenian SSR. From that POV, far better to have a Kremlin generally frustrated in its ambitions in the Straits and instead being focused on getting as much as possible on its southwestern border for lack of another place they can make gains as a result of their war with Turkey...