"Just stop caring" can mean more than one thing, and can be a very positive approach. I'm reminded of the buddhist concept of letting go of attachments, craving, aversion, expectations. It doesn't mean "I don't care", it just means not wishing, hoping, or demanding that the present moment be anything other than it is. Being open and curious about how things are and are about to become. It can be very liberating, because letting go of those things reduces the anxiety which can worsening stuttering.