The question has a contradictory presupposition:
one might care that phonemically-speaking an /ə/ followed by a rhotic
consonant in the same syllable has its pronunciation impacted
The contradiction arises from limiting this to "phonemically-speaking", but then asks about having "pronunciation impacted". Pronunciation is phonetics, phonemes are phonology. Phonemes are eventually pronounced, but that doesn't happen in the phonology. So you have to re-conceptualize the question. Let me also point out that "r-coloring" is not phonological terminology, is is a crypto-phonetic description. In phonology, we have vowels and a coda consonant /r/ (or /ɹ/ or however you want to describe it), and then we have rules of pronunciation. When you are talking about coda r phenomena, you can't talk about "English", you have to talk about a specific type of English – more fine-grained that "US English" and "UK English".
There are some purely phonological issue that relate to /r/, but they are more broadly applicable than just "after /ə/" (/a, ɔ, ɛ/ etc can also appear before coda /r/). The only thing that can be said about /əɹ/ and only /əɹ/ is that in most rhotic US dialects, it is pronounced [ɹ̩] i.e. as a syllabic rhotic without a separate [ə] transition. But that is about phonetics, i.e. how a phonemic sequence is physically pronounced. There is no phonology to that.
In phonology, r-coloring is "having /r/ in the coda", or similar prosodic position (there is controversy over the syllable position of ˈVrV). The set of vowels that can appear in that position is reduced. In my (PNW American) dialect, you get [ɪ ɛ ɔ a] and [ʊ] in sociolinguistically complex circumstances (when talking to people who speak a different dialect, and you have to say "moor", "poor", "tour"). The fact of getting [ɔ] before [ɹ] is of some interest, because in that dialect, the vowel [ɔ] only exists in pre-glide position (core, coy, coat), hence "caught" and "cot" are homophones.
Additionally, you should treat syllabic [ɹ̩] as /ər/, e.g. [bʌɾɹ̩] "butter", [bɹ̩] "burr". That treatment explains alternations like [ˈtɪɹəni ~ tɹ̩ˈɹænɪkl̩] "tyranny~tyrannical", that is, vowels reduction before /r/ yields syllabic r.
/r/ is treated more like a glide w.r.t. syllable-related consonant allophony. For example, /t/ is glottalized or even becomes [ʔ] in the coda after [j w ɹ], as in bit, beat, pout, put, putt, bite and bart but not belt (and in the case of "can't" it depends on whether you nasalize the vowel and get rid of /n/), and not after any obstruent such as "bust, pushed". This does not distinguish /r/ from the glides /j,w/.
The upshot of this is that the rhotic behaves phonologically like a glide. The one thing that is just about coda /r/ is unrounding. In this dialect, /r/ is rounded in the onset – [ɹʷæbɪt] "rabbit" – and in the coda after a round vowel ([kɔɹʷ], vs [bɪɹ, bɛɹ, baɹ]). The simplest account of this is that the phoneme is intrinsically rounded, and unrounds after an unrounded vowel. The other rounded glide, [w], does not unround in [bæw] "bow", though the only non-round vowel that can appear before [w] is [æ] which can't appear before [ɹ] – in this dialect.