I did manage to catch this, it's on Hulu! And.....personally I was not a fan, haha. The songs did seem to fit the tone of the film, which honestly was a pretty gloomy affair -- not just content, but visually as well. This resorted in the majority of the tunes being all mid/slow-tempo, with similar instrumentation, so there wasn't much to distinguish one from the other. I was especially surprised by the general lack of vocal "hooks" in the songs, which are staples of any musical, movie or otherwise -- some of them didn't even have traditional verse/chorus/verse/chorus structure, which again, is surprising given the fact they are 3-4 minute songs written ostensibly for that purpose. That does explain why the majority of them were, like Umbrellas, "dialogue-sung" -- very few times did a character use a full singing voice, which is quite rare for something that utilizes a common structure of musicals (and did make me wonder, why take the musical approach at all). Great lyrics though!
Musicals, whether on stage or film, tend to start with a song in order to set the reality of expectations for the viewer. The choice for Love Songs to just start like any other film, only to have the characters start into song without warning after things had already started rolling, was a bold choice in this sense, and one that didn't work for me, even though I knew it was coming.
And thus, funnily enough, the preferential criticism of wanting Umbrellas not be sung the entire time held true in the exact opposite fashion for me for Love Songs -- I think I would've enjoyed it much much more as a movie with no songs at all, as (unlike Umbrellas) the plot goes interesting places, and the content is handled with care. All these words probably make it seem like I disliked the film even more than I did -- I just wanted to make sure it was clear why I found the music aspect of it so disappointing!
aside: The homages to Umbrellas are absolutely there, you were spot on about that. The most obvious being the movies following the same 3-act structure, with each "part" (the departure, the absence, the return) being displayed on screen verbatim, the actress who plays Julie has her hair done up like Catherine Deneuve at the beginning (she gets a dolly shot of her own as well), and the latter's own daughter makes an appearance as a supporting character for good measure.