Your school counsellor used a bad analogy but the logic is this: every single person has something about themselves they are embarrassed by. Some are worried they are too tall, or too short, or their face looks funny, or their voice sounds odd or they laugh too much or they're too poor or too rich or they have one leg or use a wheelchair or their nose is the wrong shape....literally the list is too long to print. Truly, even those who we think are 'perfect' are in secret worried about being socially accepted because of some sort of flaw. And we become obsessed with these imagined flaws. We think more about them than others do. We become hyper-fixated that perhaps everyone else is looking at our flaws. But it's not true. We are all flawed in our own ways. We stutter, someone else has one leg, someone else is very tall for their age, someone else has an odd and embarrassing laugh... We notice our own unique traits more than others do. We also tolerate uniqueness in others more than we realise, so therefore, others tolerate our speech impairments more than we think.