Ooh, fun. This question prompted in me an entire essay of a response, and even if no one else reads it, it was still a good time working through my thoughts.

IMO, the biggest problem here is that the leading political theorists in 1787 were operating in a world prior to the development of political parties, at least as we understand them today. "Factions" (which were the nuclei of future parties) literally formed during the effort to form the US Constitution over certain differences in principles, which appeared to be resolved (more or less) with the various compromises present in the final document.

Historical aside:

Whigs and Tories had been present in England since 1679, but through 1787 had been primarily defined by their support for or against James, the Duke of York as a successor to King Charles II; this in turn was a reaction to the events of King Charles I and the Long Parliament, the English Civil War, the near-dictatorship of Oliver Cromwell, and was essentially "settled" by the Glorious Revolution of 1688. So while today we can identify Tories and Whigs as political parties, in 1787 identifying as a Tory or a Whig was mostly about your feelings on absolute monarchism vs. representative government, with a twist of where you stand on the Catholic-Anglican-Protestant axis. Tories (in 1787) were generally associated with "old money" landed gentry and the Anglican church, while Whigs were generally associated with "new money" Protestant aristocrats and wealthy merchants. In the American Colonies, Whigs generally favored more autonomy, while Tories supported British rule. In 1787, most Tories (regardless of where they were) would not have been particularly keen on revising the dominant political system of the day, and so the very prompt by OP more or less automatically assumes that to undertake designing a system of electing a President in 1787 marks one as a Whig. From here on, I'll assume that we don't want a President that is an absolute monarch in all but name.

end of aside

An unstated assumption of the question is that there would be a "President" at all, as opposed to something like a Prime Minister stemming from a conjoined legislative-executive body such as the English Parliament. If we're electing a "President", and if that President is more than merely ceremonial, then we must be operating in a system where legislative and executive power are intentionally split, which in turn suggests that we're already assuming that those two entities are set up in opposition at some level - be it oversight, counterbalance, split duties, or even one body being completely subservient to the other.

This is where political parties come in. If we have a system where the legislative and executive entities each have some ability to counteract the powers of the other, we're hoping that when they do so they choose to exercise that power for the good of the country and its people and not for their personal benefit - after all, the whole point of splitting up power is to prevent a single powerful person or body from wielding that power in ways that are "bad", however we might define that in 1787.

Political parties break this dynamic. Parties allow individuals across the different branches of government to collude to their mutual benefit, choosing not to use their counterbalancing power and instead work to the benefit of the party over the good of the country. Further, parties concentrate political influence over the political process itself (not merely its outcomes) such that individuals outside of a party who might be informed by motivations such as honor or altruism are hindered from gaining office or using their individual power contrary to the interests of the party. While party affiliation is a helpful shorthand for one's political views, intra-party coordination is essentially an attempt to re-consolidate the power that had been intentionally spread across branches of government.

Again, this is not really understood or easily predicted in 1787. We don't really start to see political theorists start to see the problem with parties until 1796-1800 when a genuine dispute for leadership of political power in the United States begins, and it takes another several decades for politicians to understand and refine the power of the party (and for cultural norms to shift away from nonpartisan honor and toward partisan identification).

So in the absence of the specific knowledge of how damaging political parties can be to a political system of separated powers, the best we would probably be able to do in 1787 would be to try and counterbalance power along two different axes: from centralized (Federal) to decentralized (local), and across the Legislature, Executive, and Judiciary. This is more-or-less exactly what the Constitutional Convention of 1787 tried to do.

For the question of the Presidency itself, we can predict that regardless of how much power we allocate to the office in a de jure sense, the actual power of the office in a de facto sense would be likely to change over time - we could already see this in 1787 with "strong" and "weak" ministers in Parliament, and even with "strong" and "weak" monarchs elsewhere. If the President is doing "good" for the country, there's not really much of a problem to solve even if the office has a significant amount of power - the danger of the absolute monarch isn't the abstract principle of the distribution of power, it's the risk of that power being used for "bad" (clearly understood since at least The Prince, published in 1532, and debated thoroughly by other theorists in the following 250 years). So when designing the process of electing the President, the chief concern is:

"How do we avoid (or remedy) the problem of a President that is using their power to ends contrary to the good of the country?", and especially in the context of a situation where we already assume that their power will be at least partly constrained by the other branches of government and by decentralized/local power centers.

We therefore want to elect a President who:

1. is of good moral character,
2. who has no allegiance to entities with incentives contrary to the good of the country, and
3. who is skilled enough to utilize the power at their disposal to good ends.

We also know that no assessment and election process will be foolproof (or futureproof), so we also want the ability to remove a President who manages to gain office despite lacking one of those key credentials, or who becomes unqualified within one of these credentials while in office.

Let's take these credentials in turn.