Let g be an odd function

Then

.

Let

given by

.

Then

, hence cannot be extended to an odd function. So your method only works when f(0) = 0 and so it reasonable to believe the span is not dense

I claim that my method does work.
But I don't know much of a difference this makes on the argument, because let's say we proceed with this constant function 1. then there is a discontinuity, does this stop us being able to talk about inner products, integrals etc?

That's exactly correct. It doesn't matter if g has a discontinuity at 0. It still has a Fourier series, which converges to g in the L^2 and L^1 norms.