An important part of language change is surely phonological variation. I'd assume that phonological changes happen involuntarily, driven mainly by articulatory mechanisms, and then slowly spread to other speakers under the influence of social factors. My question is: why aren't these phonological changes corrected when they first happen? I think that in any language, when some sort of phonological "mistake" happens , it's merely a source of amusement, but I cannot comprehend how this can turn into a permanent phonological alteration.
The main reason is that language acquisition does not operate on the basis of correction, and nothing overt in the signal indicates that a pronunciation is a mistake vs. intended. Children are exposed to data from a lot of people and use that to induce generalizations about "how people talk", so if a child grows up in a community split between Floridians, Minnesotans, and Central Valley speakers, they get "mixed" data, and the concept of "mistake" isn't applicable, at least not until/unless they are formally instructed in school about "proper pronunciation".
We can study sound changes that are already in progress, but we don't know how to catch that first moment when some change gets started. For example, the development of glottalized n in some dialects (e.g. "can't") probably depended on syllable-final non-release of voiceless consonants, but that is widely-enough distributed in English that we don't know when this first started to happen (so we didn't watch as it first emerged).
Ordinarily, sound changes are corrected. The corrections, however, consist in teaching children not to apply the changes.
For instance, many children change word final obstruents to voiceless (because voiceless ones are easier to say), but in order to acquire English, children have to stop doing that. Normal adult English does have word final voiced obstruents, so normally adult English speakers have been successfully corrected.
If at some time in the future these corrections during language acquisition are no longer effective, English will come to be like German, and word final obstruents will be devoiced even in adult speech.