Why isn't 'chemically-strengthened glass' made with potassium carbonate to begin with?
The potassium is not added first because the potassium does not intrinsically make stronger glass, it is the substitution of a larger ion for a smaller one that does.

To understand why ion exchange strengthens glass, you have to understand why the ion exchange makes the glass harder. The process of ion exchange hardening is done at a temperature that allows ion diffusion, but disallows a reconfiguration of the glass structure or relaxation of bonds (i.e. below $T_{g}$, the glass transition temperature). When sodium atoms are replaced by potassium in glass, the potassium ions occupy a site that is sized for sodium; this mismatch creates the compression that in turn gives the glass its scratch resistance. Thus the only way to chemically harden the glass is to substitute larger ions into the glass.

If you make a glass with potassium, you have potassium ions occupying potassium sites, thus no strengthening occurs.