Eventually MC does tell her, yes. If I recall correctly, you can be more open and upfront or more vague and dismissing.He does tell her. Initially he texts that he's fine after she's told he's taking a few days off, but then she calls and she's informed of Lily's condition.
And then once she finds out what's happening, she wants to get in close to support the 'most important person in her life' (I will assume) in the way she knows how. Many people in real life, if their significant other is affected by such a traumatic can be more clingy. Can't really fault her intentions for that, as she's just trying to be there for her man, in the way she knows how.
The big difference is not everyone wants/needs the same thing in trauma. Grief and tragedy play out very differently for each person. Some people need a rock for support, something like what V is (I think) trying to offer... something to hold on to, to weather the proverbial storm as it swirls around. Others turn to drugs/alcohol. And others just need some time to themselves to sort themselves out. Providing the wrong kind of support to someone in a trauma can be very abrasive, absolutely.
I don't fault her, in the story, for trying to support MC in the best way she knows how. Is it the right way? I guess that depends on the reader.