I think there's a bit of a straw man here pointing at the series definitions as being applied as the "intuitive" sense for sin and cos.

Instead, I find that the intuition that's sought is more to start by seeing exp(it) as being a generator of complex rotation---a tremendously beautiful and parsimonious bit of theory---and then seeing sin and cos as being 1-dimensional coordinate projections of that.

Then the series definitions are just cute ways of deriving that relationship formally.

Circles over triangles.