I actually never really thought this was that big of an issue but your link is really interesting and brings up some great points. Hamiltonians I think solve the largest issues but I know from talking to people who learn mechanics that that becomes confusing to because dq/dt they think should be related to momentum simply by p=m*dq/dt but it's a tad more complicated than that because you're talking really about a manifold and fancy geometry talk not commonly know to physicists