Some people make a big deal out of it, but for instance in German relative pronouns "der", "die" and "das", which all correspond to the English "that", are used exclusively for this purpose. So it is not simply that ignorant people who do not know the difference use "that" instead of "who". I suspect that it has to do with the grammar of Germanic languages in general. Of course, you can use who to make the meaning clearer or to add emphasis. Using "that", however, is perfectly acceptable and, yes, it is correct.

Some people make a big deal out of it, but for instance in German relative pronouns "der", "die" and "das", which all correspond to the English "that", are used exclusively for this purpose. So it is not simply that ignorant people who do not know the difference use "that" instead of "who". I suspect that it has to do with the grammar of Germanic languages in general. Of course, you can use "who" to make the meaning clearer or to add emphasis. Using "that", however, is perfectly acceptable and, yes, it is correct.