欠 draws a person with mouth open, breath flowing out — the radical of yawns, sighs, and songs. 哥 supplies the sound. A song is literally 'the gē that comes out on your breath'.
From the 诗经 Book of Songs — folk lyrics collected three thousand years ago — to 国歌 the national anthem, 歌 has always been how Chinese collective memory travels. Mountain folk duets (山歌) were dating apps before phones: villages courted by singing across valleys.
Big brother 哥 with his mouth open 欠 — always ready with a SONG.
The measure word for songs is 首 shǒu — 'head'. 一首歌 = one 'head' of song, the same measure word used for poems.