I think you are imagining that voting still took place in one day in early America -- which it did not. America's first election took place over a series of months. Between December 15th, 1788 and January 10th, 1789, states chose their delegates to the electoral college. Then, on February 4th, 1789 the electors gathered together and casted their votes. It wasn't even until April 6th, 1789 that Congress actually certified the votes! Then, a few weeks later on April 30th, 1789 Washington took the oath of office.

All other elections took place over weeks in the ensuing elections, with the final day of the election typically being the one where the electors gathered to cast their vote. 1792 election took place from November 2nd to December 5th, 1792. The election of 1796 took place from November 4th to December 7th, 1796. The election of 1800 was from October 31st to December 3rd. So as you can see, it took weeks, not a single day for them to figure out the votes in America's earliest days.

It's also worth noting that barely half of the states in the early republic did any type of a popular vote. For instance, in the election of 1796, only 9 of the 16 states voting had any type of a popular vote for president -- and none of those votes actually mattered since it had no bearing on who the electors would be. The national government allotted extra time for states to choose their electors and then for those people to gather together to vote for the new president.

So again, they didn't need to count a popular vote for president. States selected their own electors during the weeks leading up to the electoral college vote (usually by the state legislature), and then those folks would do the actual voting.