As a native American English speaker from the Northwest, whenever I isolate the r in words like "right" or "rope" it's always /ɚ/, the same as the r in words like "first" or "girl" but the "ir" in first and girl is considered an r-colored vowel, but the r in "right" and "rope" is not considered a vowel
Going through the checklist for vowels:
Open vocal tract - check
Voiced - check
Nucleus of a syllable - while a lot of people would say no I'd say yes
Going back to words like first and girl, you could just as easily write them as "frst" and "grl" and they'd be pronounced the exact same (in American English), the extra letter is purely convention, the r is what's forming the nucleus, and in fact in American shorthand it's quite common to remove the unnecessary letters, in British English I could see the case for r not counting as a vowel since in words like first and girl the nucleus is moreso around the i, being pronounced more like "fest" and "gel"
Just as "turn" can be shorthand-spelled "trn" in American English, but "tun" in British English, and they make the exact same sound as the actual word in their dialect, because in American English the nucleus of the syllable is "r" but in British English the nucleus of the syllable is "u"
So if the American r fulfills all the requirements of a vowel, why isn't it considered one?