But don’t we also observe mourning during sefirah on the national level in addition to Tishah BeAv, and it also falls out during Nissan?
I’ve never seen this issue addressed, but two distinctions come to mind.
1) The mourning of sefirah is not a single day, it’s drawn out over a long period.
2) The tragedy which we commemorate is different than that of the churban; it’s the tragedy of the Torah lost upon the death of R’ Akiva’s talmidim.
It’s also likely that chachamim who instituted the mourning during sefirah understood that tragedy to be independent of the churban, whereas the other tragedies, as R’ Moshe writes, are an extension of the churban.
The idea that Nissan is a time of happiness would seem to be a support for those who keep the halachos of mourning after Rosh Chodesh.