The quality of the US officer corps would suffer somewhat, but there were states with military institutions to train their militias, and the 'old aristocracy' would see to it their children got a thorough grounding in military affairs, West Point or no West Point. We might see more of them going to Europe to study military arts - probably in France, though possibly in Prussia and Britain, too.
An interesting side effect would be the state of engineering in the early USA. The country had anb overspill of highly qualified civil engineers who had graduated from West Point (one of the most technical academies of its time in its emphasis on artillery, fortification and military engineering - since the country already had a surfeit of good horsemen and brave riflemen). The army simply couldn't absorb all the graduates, so a lot of them found themselves, willy-nilly, planning towns, building bridges, running railways or marking channels. Without this steady supply of quality personnel, US industrialisation and infrastructure developmnent would probably suffer and the private sector (or concerned state governments) might choose much earlier to establish and fund technical colleges.