I agree. I had an SMS growing up and always noticed the music sounded "cheaper" than the NES, almost childish. I think it really was just the square waves making everything sound the same. The NES had more interesting output with its triangle and sawtooth wave output and it gave it more edge and character.
It may not have had a sawtooth but it did have the DMC (sample channel) which although very quirky could create a lot of variety - and used melodically to give you, for instance, a sampled bass - or drums - or an orchestral hit!
The NES' own sound chip didn't have a sawtooth channel, but some games had an onboard sound chip that added one, like Konami's VRC6: https://www.nesdev.org/wiki/VRC6_audio
