General Order One forbids contact with a civilization or culture that has yet to achieve warp speed in order to avoid cultural contamination and alter the natural development of a society.
While the Humans that the Discovery found were not an independent species but were abducted from Earth, they did come from a time when warp speed had yet to be invented on Earth, and was thought of as a flight of fancy at the time, and, since their abduction, they formed a new society and a new culture which had not developed warp speed either.
Thus why the Prime Directive was applicable to them: neither those Humans nor their ancestors had access to warp technology at any point.
Besides, there have numerous examples throughout the various series of the deleterious effect of First Contact with a pre-warp society:
- In the aptly named First Contact episode of TNG, a high-level government official commit suicide and tries to frame Commander Riker from it, because he was so afraid of his planet being invaded by the technologically superior aliens.
- In the TOS episode A Private Little War, Klingons provide a primitive society with advanced weaponry, disrupting the status quo and giving one faction the means to overpower and defeat their opponents, forcing Kirk to intervene.
- In the TOS episode A Piece Of The Action, a book titled Chicago Mobs of the Twenties is inadvertently left behind on a pre-warp planet, introducing radical ideas and leading to a complete change of the society in a century.
- In the TNG episode Who Watches The Watchers, Picard and the Federation observers are deemed as deities by the proto-Vulcans on the planet, nearly prompting them to abandon their nascent logical ways and go back to the old religious ways, which included sacrifices to the deities, putting Commander Troi's life in jeopardy.
Warp technology is a useful metric for contact because, at that point, the civilization has the means to go out and establish contact by themselves, which it will given how full of life and civilization the Milky Way is, and they're supposed to be more or less ready for it.
As a side note, by the time of Discovery, Earth has united under one single government and everyone on the planet enjoy access to the same resources and amenities. There are effectively no 'less-informed poor society' left on Earth, except for the people who chose to forbid the available technology.