They put Reigen's character song on spotify and it's basically just Reigen speaking over music with a choir singing the chorus so. The easy conclusion to gather is Reigen can't sing
Takahiro Sakurai has sung other character songs so this is clearly a choice for the character. Reigen doesn't seem like he'd be embarrassed of singing if he was good at it so we can assume he's just tone deaf
The elves would have to invoke it, though.