Do /ɪə/, /eə/, /ʊə/ diphthongs actually exist in General American as phonemes?

No, they do not.

The diphthongs /ɪə, eə, ʊə/ found in Received Pronunciation stem historically from the sequences of /iː/, /eː/ (now /eɪ/), or /uː/ + /r/. In about the 16th century, /iːr/ etc. started to become realized more like [iːər] etc. Later they became more like [ɪər] etc. At this point it would be unreasonable to posit the diphthongs as phonemes because /iː/ etc. were consistently realized as diphthongs before /r/ (so the diphthongs were merely allophones). But then it became fashionable to drop /r/ after vowels in Britain. This gave rise to the diphthongs /ɪə, eə, ʊə/ as distinct phonemes in British English.

General American didn't go through the r-dropping, so even if the diphthongal realizations were retained (which they kind of are—see below), it would be more theoretically sound to analyze the diphthongs as allophones of the long vowels which they originate from. But in GA, /iː, eɪ, uː/ and /ɪ, e, ʊ/ are neutralized before /r/, meaning there are no minimal pairs of /iːr/ etc. and /ɪr/ etc., in that nearer and mirror rhyme and so do Mary and merry (which they do not in RP). This is why some phonemicize near as /niːr/ and others as /nɪr/. (Often the length mark <ː> is omitted and <ɛ> substitutes for <e> in analyses of AmE, but here they are retained for comparison with BrE.)

/niːr/ and /nɪr/ are both defensible, but the latter is more common because key-ring and hearing typically don't rhyme (some pronounce zero, hero, etc. with /iː/, but these are rare exceptions and in free variation with /ɪ/). And although the vowel is short and monophthongal when followed by a vowel, as in caring or nearer, when preceded by a pause or consonant, as in cared or near, the vowel is often diphthongal and resembles the RP counterpart (except followed by /r/). But that doesn't mean they are phonemes in GA, which requires minimal pairs. (Also note marry /ær/ also merges with Marry, merry /er/.)

See Wells's Accents of English (1982), vol. 1, sections 3.2.1 and 3.3.1 for more.

Those who insist on the existence of centering diphthongs as phonemes in AmE do so either because they think that allows for better comparison with RP, because their model accent is different from what linguists have referred to as "General American", "Network English", etc., or because they don't know what they're talking about.