Yeah, in analyzing rhotic North American speech I consider them to be distinct phonemes, /ɑɚ/, /ɛɚ/, /ɪɚ/, /ɔɚ/ and /ʊɚ/ (along with /ɚː/. I have an interesting perspective because I speak a GenAm-ENE hybrid dialect, so I'm rhotic but make a lot of the pre-/ɹ/ distintions associated with non-rhotic speech. In my case at least, these rhotic phonemes are indispensible, because the sounds in question are different from both simple lax vowel + /ɹ/ sequences (Mary [ˈmɛɚi] ≠ merry [ˈmɛɹi]) and tense vowel + /ɹ/ sequences (jury [ˈdʒʊɚi] ≠ Jewry [ˈdʒuːɹi]; see also [ˈmoʊɹi]
, one possible pronunciation of "Maori"). Cases of the latter seem to be vanishingly rare, but I do find them to be phonotactically allowable: I'd analogize them to the similarly rare [ɑːw] sequence (≠[aʊ]) that I use in words like "Kurosawa" or "Peshawar". But anyway, the crucial thing is that the rhotic sound is inseparable from the vocalic sound that precedes it: I can split "merry" into "meh-ree", but "Mary" only into "mare-ee".
Now a lot of sources will tell you that Northeasterners like me have special "extra" phonemes in words like "mare/Mary" and "near/nearer", which are lacked by those in the rest of the country – but I think this is a backwards way of analyzing things. When I hear people from most of the rest of the country speaking, I don't usually notice any difference between their "mare" and "near" (or "Mary" and "nearer") and mine – what I notice instead is that they seem to be using those sounds everywhere. GenAm "merry, ferry", "mirror, Sirius" and (especially) "worry" sound to me like "Mary, fairy", "merer, serious" and "whirry" (and other people who I've spoken to seem to agree on this point too). So I think what's happening isn't that we distinguishers have more phonemes, but rather that distinguishers allow sequences of lax vowels + /ɹ/, whereas in merging speech these sequences are disallowed and collapse into the nearest free rhotic phoneme.Merriam-Webster
, despite its non-standard notation, does essentially agree with my approach: for "ferry" they give \ˈfer-ē, ˈfe-rē\ – merging and distinguishing, respectively – whereas for "fairy" they give only \ˈfer-ē\. (Even though their general practice is to assign consonants to the following syllable wherever possible: for example, for "messy" they give \ˈme-sē\.) Likewise, for "hurry" they give \ˈhər-ē, ˈhə-rē\; for "furry" only \ˈfər-ē\. (I'd note that even sources that take the misguided approach I mentioned above often still end up positing a distinct r-bound phoneme in "tore, Tory". They also seem more amenable to positing a distinct r-bound phoneme in "fur, furry", perhaps because the phonetic gulf between [ɚ] and [ʌ] is so great.)
Now I'll admit that some of the strength of this approach is that it makes interdialectal comparisons nicer: if we were dealing with a world with only mergers, then analyzing their pronunciations as sequences of lax vowels + /ɹ/ would be less objectionable. But I still think that the rhotic element tends to combine with preceding vowels in a distinct way, and that a typical GenAm speaker, if asked (as a phonetic exercise) to produce "meh-r" or "meh-ree", wouldn't find those to be identical to "mare" and "Mary/merry" in the way that they would find "meh-ss" or "meh-see" to be identical to "mess" and "messy". My impression is that they act just like diphthongs, and that trying to split them into diphonemic sequences would be about as fruitful as trying to analyze /aɪ/, /aʊ/, /ɔɪ/ as /ɑː/+/j/, /ɑː/+/w/, /ɔː/+/j/.
In Scottish English, yes, I'd analyze them as having sequences of vowel phonemes + /r/. The phonemics and phonetics of Scottish English are quite distinct from those of other dialects, retaining many elements from Scots (which is arguably a distinct language). The realization of /r/ as [ɾ] helps it to act more like a "normal" consonant, rather than blending with nearby vowels as we see in other dialects.
And remember, my friends, future events such as these will affect you in the future.