I think your analysis is reasonable. Why else would he suddenly send you a song? He sent you a song because he wanted you to listen to the melody or enjoy the beautiful melody of a song with you, which could bring you a good mood or help you solve problems. In short, he wants to make your mood better, or the meaning of the lyrics inside. You can experience it, or you can just say it. But sending a song can't decide too many things, and you don't have to think about it, and you don't have to analyze it too carefully and sensitively, which may not play a good role. Think of it as enjoying a song from the past. Isn't this a better thing?