British people are taught that their local dialects / pronunciations are something to be proud of
This is the opposite of my observation. I have known quite a few from across the pond who had it hammered into them in grammar school and beyond that their accent and dialect were things to be ashamed of mightily.
Trying to describe how a word is pronounced by resorting to rhymes is a less than optimal solution. For me, cot and caught to not rhyme, but for many they do. Same with pin and pen (for me). The best we can hope for in transcribing pronunciations is to use the IPA in conjunction with a detailed phonology of the language / dialect in question. All the rest leads to madness.
My dialect has both a schwa (e.g., sofa /'soʊfə/) and an r-colored schwa (e.g., finger /'fɪŋgɚ/. They are quite distinct. Cf. the popular transcriptions of US um and UK erm which are both basically /əm/.
