Suddenly everyone’s a critic!
Seriously though, aside from leaving the sanctuary, I’m not sure he’s necessarily doing anything wrong -
I’m not aware of anything in the GRIM which prohibits the singing of a homily. the problem with making judgments about what is and isn’t supposed to happen at mass is that, within the boundaries of the GIRM, there’s a fair amount of scope and that’s why not all masses are the same - and indeed, never will be, because the people involved aren’t all the same. I lived in a parish once which managed to cater for a variety of liturgical tastes across its Sunday masses - from folkstyle, to family, to high church there was something for almost everyone!
In fairness, I’d prefer my priest didn’t sing his homilies and not just because I’m not sure that they’d go too well with his singing voice! At the same time though, I’m only too aware of the humanness and fallibility of our priests and so would suggest to the OP that they (gently) approach the priest with their concerns, explaining how you feel rather than how you think things should be. After all, an overly enthusiastic priest (and I’ve met a few of those before) doesn’t need to be a cause for complaint but instead a source of energy to be harnessed.