/ɪə/, /eə/, /ʊə/ as phonemes?

There is a contrast (in most non-rhotic varieties of English) between words like "ferry" /feri/ and "fairy" /feəri/. How should we analyse them if /eə/ = /er/? As /feri/ and /ferri/? That doesn't seem right to most people. Perhaps you could use syllabification (/fe.ri/ vs. /fer.i/), but people don't really agree about how to syllabify words like "ferry."

Another reason for the r-less notation is for parallelism with the notation of two other phonemes that can correspond to rhotic vowels in rhotic Englishes: /ɑː/ (START lexical set) and /ɔː/ (NORTH and FORCE lexical sets).

Since /ɑː/ and /ɔː/ in non-rhotic dialects also have other origins than vowels historically followed by r (the BATH and PALM lexical sets, and the THOUGHT lexical set), we can't analyze them as /ɑr/ and /ɔr/, unless you want to say that words like bath [bɑːθ] and caught [kɔːt] have somehow undergone a phonological shift that added the phoneme /r/ after the vowel. (Most phonologists do not want to say this.)

The sounds [ɪə] and [ʊə] can also arise in some dialects from sequences that lack historical /r/ by way of vowel coalescence (in words like idea), so the same issue of historical development applies there.

In most American dialects, /ɪər/, /eər/, /ʊər/ are allophones of /ɪr/, /er/, /ʊr/. Some speakers say /nɪər/ (near), but /nɪrər/ (nearer) and /mɪrər/ (mirror). So if you speak one of these dialects (or are learning English from somebody who does), this distinction may be confusing.

Dictionaries make these distinctions because they exist for many speakers. In the U.S. Northeast, many people (both rhotic and non-rhotic speakers) still make these distinctions, and outside of North America, they are hardly ever allophones.

From what I understand on phonetics/phonology, /ɪə/, /eə/, /ʊə/ can simply be considered as allophones of /ɪr/, /er/, /ʊr/

I'm not sure where you derived this understanding from. It seems obviously incorrect as those sets are distinct by definition of the symbols themselves. I quote:

Since phonemes are abstractions of speech sounds, not the sounds themselves, they have no direct phonetic transcription. Allophone - Wikipedia

Maybe you are referring to non-rhotic* varieties of English (such as the one I speak). For example I pronounce 'iron' and 'ion' identically.

However in the US for example those two words are quite distinct. They are also different in Scotland, and some regions of England.

So, if you are learning the Received Pronunciation* version of British English you have to adjust your understanding of the usage of the symbols but not their meaning. An American dictionary will not tell you how to pronounce RP English.

Rhoticity in English - Wikipedia Rhoticity in English refers to the situations in which English speakers pronounce the historical rhotic consonant /r/, and is one of the most prominent distinctions by which English varieties can be classified. In rhotic varieties of English, speakers pronounce /r/ in all instances, while in non-rhotic varieties, speakers no longer pronounce /r/ in settings in which it is not followed by a vowel

Received Pronunciation - Wikipedia Received Pronunciation (RP) /rɨˈsiːvd prəˌnʌnsiˈeɪʃən/ is the accent of Standard English in the United Kingdom, with a relationship to regional accents similar to the relationship in other European languages between their standard varieties and their regional forms.