There are various things that might be denied. One is that humans have a symbolic system for categorizing speech. I take that to not be reasonably deniable, however I believe that certain phoneticians adhere(d) to a non-categorial analysis of speech recognition.
Another that can be denied is the specific package-deal "phoneme" as contrasted with allophone with all of that biuniqueness stuff. I do deny that there is such a thing as a "phoneme", where /t/ is a phoneme of English and [tʰ t̚ ɾ] are allophones. It is true that [tʰ] can be derived by rule from /t/, but that doesn't justify creating a special theoretical category. There is a difficult question as to which structures exist in the phonological component and which only exist in the phonetic component, and for example there is good evidence that "nasal vowels" are not a phonological category of English, they are the result of phonetic implementation: but the flap is clearly a phonological entity.
A further question is what kind of thing a "phoneme" is. Is it part of phonological grammar, phonetic grammar, or something on the periphery of language, at the auditory interface with phonetics? The "speaker awareness" test is, indeed, not reliable exactly because speakers need specific training in their writing system, and they do not learn a general skill about "phonemes" from one language that can be translated to other languages. E.g. a speaker of Logoori may know how to write English and Swahili, and even Logoori but the writing system is defective in not representing all of the phonemes. Unless one engages a speaker in training in linguistics, the standard reaction to a vowel-quality, vowel-length or tone minimal pair is that "they are spelled the same". You can experimentally get them to identify words, but they don't have any awareness of what's a "phoneme" in their language.