As I put it to Michael,
there is in each case also an optional variant involving schwa plus a nonsyllabic consonant.— to which he replied that he didn’t know what I meant.
I’m not sure how to put it more clearly. I mean that although the word hidden, for example, is mostly pronounced ˈhɪdn̩, it can also be said as ˈhɪdən. Most cases of n̩ can be replaced by ən, and vice versa, with no change of meaning. And the same applies to the other syllabic consonants of English. You can say əl instead of l̩ in medal - meddle (though that might sound odd or childish, depending on where you come from). For hesitant you can say ˈhezɪtənt or ˈhezɪtn̩t. For blossom you can say ˈblɒsəm or ˈblɒsm̩. For gathering you can say ˈɡæðərɪŋ or ˈɡæðr̩ɪŋ (= ˈɡæðɚɪŋ), or indeed compressed as ˈɡæðrɪŋ.
In terms of phonology, I would say that syllabic consonants are not phonemes, i.e. not part of our underlying sound system. Rather, they are derived by rule from an underlying string of ə plus a non-syllabic sonorant consonant. I call the rule Syllabic Consonant Formation, and it takes the general form
Two segments are reduced to one, with the sonorant consonant retaining its various attributes (place, nasality/laterality, etc) as it acquires syllabicity.
The conditioning environment of the rule (shown here just as “…”) is pretty complex. It varies according to different accents and different speaking styles, and also depending on which consonant is concerned. For ən after a strong vowel plus d, as in garden, the rule is strongly favoured (though evidently now becoming less so in some BrE). With a preceding fricative, as in lesson, it is still favoured, though perhaps less strongly. With an affricate, as in kitchen, it is disfavoured. In common and lion, i.e. after a nasal or a vowel, it is so strongly disfavoured as to be virtually unknown in RP-style English. Although a syllabic nasal following a nasal is a no-no, a syllabic lateral, on the other hand, is fine: channel ˈtʃænl̩.
Although the AmE NURSE vowel could in principle be analysed as a strong (= stressable) syllabic r̩, this would not fit the above rule, which requires a weak ə as part of the input. So I treat the NURSE vowel in both BrE and AmE as a primitive, ɜː ~ ɝː. The second vowel of AmE father, however, does fit, and I analyse it accordingly: ˈfɑːðər → ˈfɑːðɚ.
This is the reasoning behind the notation I use in LPD, where potential syllabic consonants are shown either as əl ən ər əm or as əl ən ər əm, depending on whether a syllabic consonant is more or less likely as the output. The LPD notational convention is that a raised symbol denotes a possible insertion, an italic symbol a possible omission. So ən implies a default n̩, as in hidden ˈhɪd ən → ˈhɪdn̩, while ən implies a default ən, as in hesitant ˈhez ɪt ənt → ˈhezɪtənt.
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Here by request is a quick-and-dirty video of me saying ˈhɪdn̩ ˈhɪdən ˈmedl̩ ˈmedəl. Sorry about the poor sound quality.