Integrating a triangle wave does get pretty close to a sine wave. But the main problem with using integrators is that the gain of the output will be inversely proportional to the frequency. If you want to vary frequency without changing amplitude, then that won't work (unless you put some automatic gain control in somewhere, which would be a big step up in complexity).
And using filters on a square or triangle wave has issues too when varying frequency, since your filters will probably have fixed characteristics (unless you again have very sophisticated automatic controls).