[FiatLuxDave]

Actually, you only feel it if the force is applied differentially (which most forces we experience are). If your body had a volumetrically uniform electrostatic charge, which was not high enough to have charge screening, and you were acted upon by an electrostatic force, you wouldn't feel anything either. It is just that 'mass charge' is always distributed evenly along with mass, while with electrostatic forces this is a special case.

Forces can indeed be modeled as a curvature in space-time. However, there is a bit more to GR than just that, which is why it took Einstein years to go from SR to GR.