First of all, being sad and being depressed are two different things. One who is sad is motivated to, if at all possible, remove the source of the sadness, or at least grow from it and become a better person. One who is depressed tends to wallow unproductively in that depression. Now we are very sad. But we are not depressed.
Secondly, I heard (unconfirmed) that R’ Yaakov Kaminetsky ZT”L stopped the band at one of his grandchildren’s Chasunos from playing this song, since it was not Emes. I have heard a “Litvishe” alternate version that goes, “Simchah Gedolah Liyehos b’Mitzvah Tamid”.