I try to find an answer by self-experimentation.
I use the following contrasting examples:
Fahrt /[faːɐ̯t]/, leer /[leːɐ̯]/, Bär /[bɛːɐ̯]/, wir /[viːɐ̯]/, Moor /[moːɐ̯]/, Gehör /[ɡəˈhøːɐ̯]/, Uhr /[uːɐ̯]/, Kür /[kyːɐ̯]/
dort /[dɔʁt]/, wird /[vɪʁt]/
rot /[ʀoːt]/, drei /[dʀaɪ̯]/, Fahrer /[ˈfaːʀɐ]/, Lehrer /[ˈleːʀɐ]/
In my own speech organs, I find when practising the following:
ɐ̯ is just a vowel, with a "fixed" position of all parts of the mouth (no movement, you can pronounce it perpetually)
ʁ has a movement towards tightening the throat, from ɐ̯ towards ʀ.
ʀ again is a "fixed" sound, you can pronounce it over a lenghty time, perpetually. It is pronounced with the throat (sorry, I don't know the exact names of those parts of the throat there behind) very tight, so that it has a 'grinding' sound. Well it is a frikkative anyway, however, it is also sonant: you do not produce only the friction sound, but you add sound coming from your vocal cords.
I do not guarantee that phoneticists would agree, but that's what I seem to be finding in my own pronunciation.