It goes without saying that describing a Bioware game as a dating sim is hyperbole, although DAII did get overtaken somewhat by the romances.
In Mass Effect, the opposite occurred. The romances were nowhere near as central as the tone of the relationship suggested they should be.
I must have played some crazy East European version of ME then.
In ME2 and ME3, you had 5-10 unique dialogues per character and had one specific shot in one specific conversation to get on a romance track (whereas the DAII conversations were populated with Flirt options). The romances only rarely blended into the central narrative, but the heavy tone and stylized treatment of each romance suggested they had a meaningful role in Shepard's journey.
DAII was a more extreme version of ME2 in that the 'central narrative' was substantially reduced in favor of giving each companion their personalized arc. The romance permeated these to a much stronger degree than in ME2, which is what imparts the sense that romance overtook DAII.
It would be more accurate to say, "Companions overtook DAII, and then romance overtook companions."