This is all fairly confusing.
Countries that elect their chief executive currently almost always do so by nationwide popular vote, with a majority requirement, and a run-off (or alternative voting) if no candidate wins a majority in the first round. Using the nationwide popular vote without a runoff (meaning, using plurality voting), is rare, I think as of 2020 only one country did that. When used, electoral colleges are based on the legislatures, often assemblies of regional legislatures, and there is no attempt made to consult the electorate.
The American electoral college system is really sui generis. To be fair to the Americans, in 1787, only two important countries, Poland and the Netherlands, elected their heads of state, so the Americans didn't have much to go by. Both the Polish and the Dutch systems were political disasters that expired before the 12th Amendment.
So I will interpret the question as to whether the Americans could have gotten the Electoral College and some sort of run-off, without just adopting a nationwide popular vote with a runoff like most 20th century republics. There are two ways to do this, both possible.
The first would have been for the 1787 constitution, or the 12th Amendment, to provide that no candidate got an Electoral College majority, the electoral college itself would vote among the top two. I'm not even sure how this would have worked under the weird original system where each elector got two votes, and the top two vote getters would be the President and the Vice President, but then factions ran tickets for the two offices, so in one election the losing Presidential candidate became Vice President instead of the running mate of the winning faction, and in the second case the winning faction had both their candidates tied. Under the 12th Amendment, this would have been doable. It would have changed potentially the election of 1824,the only time no candidate got an Electoral College majority, if the Crawford electors had voted for Jackson. But its hard to see this happening with the Electors not assembling as a body, but voting at their state capitals, sometimes using the secret ballot.
A second method would be for the states to provide that their electors be chosen by votes of the electorates -this is purely a matter of state law- and that furthermore if no candidate within the state got a majority of the state vote, there would be a runoff of the top two candidates later. I don't think this is what the OP is asking for. No state has done this, and it has not to my knowledge been seriously considered. Given how these things work out, once a certain amount of states did this, all but a couple of the states would do this. It would lead to interesting elections after the main elections. Some states do provide a majoritarian requirement for their statewide elected officials, but only a minority do this.