First, the premise of the question is wrong. Strictly speaking, the Electoral College decides who is to be president, not the public vote. This is dealt with in depth in many other questions on this site, starting with "Why was the Electoral College the system selected by the founding fathers?", so I will leave reading about it as an exercise and not deal with it in depth here.
The impeachment system used by the U.S. Constitution works the way that it does because it was taken from another system that worked this way.
It is taken from the U.K. Parliament (which I will use to refer to the Parliaments of the U.K.'s predecessor states), which has a system of this form and has had for almost six and a half centuries, since 1376. (The last impeachment in the U.K. Parliament was in 1848, but it has not necessarily fallen into desuetude for being unused for 171 years. There was a 162 year gap in impeachments between 1459 and 1621, whereupon Parliament merrily started using it again, with barely a blush of embarrassment.)
In the U.K. Parliament, the lower house (the House of Commons) passes a resolution and sends one of its members to formally inform the upper house (the House of Lords) that an impeachment has been made, and (then) the lower house approves and delivers Articles of Impeachment to the upper. The Lords holds a trial, with managers sent from the Commons to conduct it. The accused can present evidence and summon witnesses, and the Commons managers demand a ruling from the Lords upon their Articles. The Lords, operating with a special presiding officer if a Lord is themself on trial, either dismiss or pass the Articles by a vote. If passed, the Commons then has the opportunity, which it can decline, to demand judgement be rendered upon the accused.
Much of this may seem very familiar to those looking at U.S. impeachment proceedings.
However, like some other aspects of the U.S. Constitution, the Framers constructed it their way in order to curb some of the egregious excesses of U.K. history. Some examples should make this clear:
- In the U.K., pretty much anyone, except a member of the House of Commons (the Commons holding that its members are privileged against impeachment), can be impeached. In the U.S., only civil officers of the United States, and the President and Vice-President, may be impeached.
- In the U.K., the impeachment proceedings are a criminal trial, and judgement has extended in some cases to banishment and sometimes even execution. In the U.S., the only allowed outcome is removal from office, and optionally a bar to holding office again. Criminal matters are required to go through the criminal courts, with all of the ordinary rights of the accused in criminal matters in place.
- In the U.K., even members of the House of Lords itself have been impeached. In the U.S., the procedure for removing a Senator is expulsion, not impeachment.
- In the U.K., some people languished for years in prison awaiting the Lords to schedule their impeachment trials. At the start of the Parliament under James 2 in 1685, there were petitions from three Lords who had had impeachments hanging over them, untried, since 1678. In the U.S., although the Constitution is mute on the subject of rights in an impeachment trial, Amendment 6 only talking about criminal trials, the general political principle that people have a right to a speedy trial would no doubt be applied far more vigorously.
- In the U.K., although the Crown could not prevent impeachment proceedings from happening by issuing a preëmptive pardon, which was attempted in the case of Lord Danby, it was held that the Crown could afterwards pardon a person who had been judged guilty. The Commons also has what amounts to a pardon power, as it has held that the Commons can decide not to request judgment from the Lords, and the Lords may not proceed to judgment on its own initiative. In the U.S., the Present is denied the ability to pardon impeachment convictions.