Now imagine plopping a foundling in the middle of it all.
Mind you, in Tom Riddle's day they didn't seem so interested, since no one knew until he tracked the information down, and even then he was able to pass himself off as a pureblood to Death Eaters. But if they cared. . . .
Some would no doubt say that the foundling was obviously not of a magical line, since the child had just been dropped off. Or perhaps claim it was a desperate attempt to hide inferior bloodlines by pretending a foundling's origin was not known. (During WWII, the Nazis quickly guessed that all claimed foundlings were Jewish infants.) Others would no doubt take the tack that (assuming a society where people didn't abandon infants for hunger) there was probably a wizard feud behind it and a desperate effort to protect the child from foes; what sort of muggles faced that danger?
Take a world, say, in which the Hogwarts founders had agreed to teach only pure-bloods or perhaps half-bloods. That wouldn't prevent muggle-born. Some of them would learn to use it. Things could get very ugly indeed in the conflict. (And the purebloods would no doubt point to muggle-born violations of secrecy as proof that wild magical talents doesn't come with knowing how to act.)