Is it problematic to listen to music during the year because of zecher lichurbon?
The Gemara in Gittin (7b) indeed writes that it is forbidden to listen to music the whole year due to us being in a constant state of mourning over the churban. The Rambam (תענית פרק ה הל’ י”ד) rules that it is forbidden to listen to musical instruments at any time, and it is forbidden to sing by a banquet. The Shulchan Aruch (או”ח סי’ תקס סעיף ג) rules like the Rambam. However, the Rama (שם) writes that music is only forbidden to one who accustoms themselves to listen to music consistently. The Bach (שם) rules that all music is forbidden, whether from an instrument or singing, even if it is not by a banquet. This opinion is quoted by many acharonim including the Magen Avraham (סי’ תקס ס”ק ח), Shlah (תענית אות יד), Mishna Berura (ס”ק יג) and many others.
Even so, it would seem that common practice is indeed to listen to music year round. There are two reasons that one can be listen to music. The first reason is that without listening to music many people would become depressed. This is first written by the Maharshag (חלק ב סי’ קכה), and brought down by the Chelkas Yaacov (או”ח סי’ סד) and Rav Shlomo Zalman Auerbach (עליהו לא יבול ח”א עמ’ קצח). The second reason to allow music is if the music is for a mitzva (רמ”א שם). The Shevet Halevi (ח”ו סי’ סט) writes that a kumzitz that brings spiritual inspiration is included in this. Rav Moshe Feinstein (או”ח ח”א סוף סי’ קס”ו) writes that a dinner for the purpose of raising money for tzeddaka or a similar event is a mitzva, therefore one may play music by such an event to help people enjoy themselves and donate more to the cause.
Tl;dr: Music is permitted to prevent depression or for a mitzva, such as a tzedakka event or a kumzitz that brings one closer to Hashem.