You are correct and that is what the original United States Constitution expressed (actually it was even more permissive; it makes no mention of parties at all. Parties could choose to hold a nomination or none at all. This has always been beyond the government's prerogative to control). Prohibitions against partisan control were significantly weakened with the 12th Amendment in 1804, since which time there have begun to be general elections in which the public votes for presidential candidates and States have thenceforth been in the business of propping up candidates to appease the national parties, and now even the amended version of the Constitution is ignored as partisan-government incest abounds.
Because the political parties now have nationalized oversight over how elections are even conducted, and how votes are allowed to be allocated. This is evident in the introduction of bound delegates per partisan rules, the existence of unconstitutional partisan "tickets" formed at national party conventions, and the existence of general elections by the public for presidentpresidential candidates. The Constitution allows none of this.
- There were no general elections for presidentThe public did not vote for presidential candidates. States appointed electors. Parties existed, but electors voted for the president in unbound fashion.
- Electors could vote for two candidates and the runner-up became vice-president. This eliminated the problem of being run by a two-party system since any administration would almost of necessity consist of party-heterogenous candidates. This also meant third parties actually stood a chance whereas since 1804 they have not.
- There was no national popular vote (writing this for posterity to remind them of the way things were).
Under the original Republic, State legislatures determined how to appoint each State's electors, independent of all other power mechanisms. They could have done a statewide popular vote, a straw poll, or had their legislators nominate and choose electors, held nonpartisan caucuses, or anything else allowed under the Constitution. Once those electors were chosen, their own judgment was the test for who became the next president and vice president. If you voted in a State popular election for your elector(s), your elector(s) would bear your representation and vote according to his/their own judgment and conscience. That would be all you had to do with any presidential election, unless you had further involvement with a national party's nomination process. There was no presidential primary and no general election by the public for the office of president. The only election you would normally vote in was to choose your elector(s), if your State legislature had invited your participation in that electoral process. If and only if you were an elector, then you could vote for two presidential candidates, the runner-up of which would be vice president.