I can speak a bit about how this "came to be": the American colonies each had their own individual voting laws and their own voting restrictions. In this, the colonies were really just following the practice established in Britain, where elections were, for the most part, locally controlled by each separate electoral district.

Let me give you a few examples of the variety of voting laws in the American colonies. In most colonies, voting was an extremely public act, accomplished by open voting, or voice voting. So if you were to cast a vote in Virginia in the eighteenth century, you would generally climb up to a polling platform in the center of town, stand in front of an official called a returning officer, and declare your choice in front of the entire community. Open voting was considered by most colonists to be the fairest system: it prevented electoral fraud (stuffing the ballot), and it also meant that freeholders did not necessarily need to know how to write in order to cast their vote. However, the colony of Pennsylvania was an early adopter of the written ballot--the system we are familiar with today. This was partly due to the influence of Pennsylvania's significant Quaker population, who believed that voting was a matter of conscience and should be exercised free of outside pressure.

Voting qualifications also differed significantly from colony to colony, and often even from district to district. For instance, there were usually property qualifications to vote (the amount was variable, but about 50-100 acres). However, in certain colonial cities like Albany and New York City, white men could also qualify to vote by paying a small fee to register as a "freeman" of the city. As I mentioned earlier, the colonists were patterning their electoral processes pretty directly after Britain, and this was true with voting qualifications as well. The city of London also allowed its white male inhabitants to register as freeman. So London working-class artisans and tradesmen voted in fairly high numbers, leading some historians like Penelope Corfield to call it a "proto-democracy."

By the time of the writing of the Constitution, the default option was to continue to allow each locality to control its own election rules and practices, as they had already been doing for centuries. And that is what was essentially written into the Constitution in Article 1, Section 4: "the times, places, and manner of holding elections for Senators and Representatives, shall be prescribed in each state." However, the Constitution also gave Congress the power to alter those regulations if it saw fit. I do not know of any serious discussion to impose one uniform election policy on the entire United States: that idea would have gone against centuries of British and colonial precedent that had established elections as an extremely variable, and extremely local process.

Sources:

Richard Beeman, The Varieties of Political Experience in Eighteenth-Century America (2006)

EM Green, PJ Corfield, C Harvey, Elections in Metropolitan London, 1700-1850 (2014)