The short answer is that some people do pronounce the vowel in sing, king, think, zinc etc. as [iŋ], but some people pronounce it as [ɪŋ]. Because there is no possible contrast between [i] and [ɪ] in this context in English, it's hard for most English speakers to hear the difference. A lack of contrast between otherwise contrastive sounds is called "neutralization". Linguists can usually argue that the sound is identified with one phoneme or anther (phonemes, transcribed in slashes, represent the grouping of sounds that count as "the same sound" for the sake of a language's sound system), but it isn't always clear. This appears to be an unclear situation: these words arguably contain the phoneme /i/ for some speakers (like you and some of the authors of posts linked below), but /ɪ/ for others (like me and James K).
Actually, some pronunciations may be in between, and some speakers might hear an in-between sound as the phoneme /ɪ/ while others might hear the same phonetic sound as the phoneme /i/. This also happens for the neutralized high front vowel sound found before /r/ in words like fear, near: some American English speakers think it sounds more like /ɪ/ and others think it sounds more like /i/. See the following post for more information about the near vowel, which I think is a very comparable situation: -eer vowel (accent/dialect variation?)
So while your experience of never hearing [ɪŋ] is not necessarily wrong, it's not highly trustworthy evidence either: some examples that you would hear as [iŋ] might actually contain an in-between sound that another speaker might hear as [ɪŋ]. Furthermore, the English-speaking world is a big place. Personally, I can hardly ever hear a noticeable distinction between the cot vowel [ɑ] and the caught vowel [ɔ] when listening to another American English speaker, but based on research that I have seen, many American English speakers do in fact pronounce these words with distinct vowel sounds: I either do not notice it, or the speakers in my surroundings don't have a distinction, but speakers in other places do. Something similar may apply for you with [ɪŋ] vs. [iŋ].
Some conventions for phonemic transcriptions in dictionaries were based on the pronunciation of specific old accents
Most likely, the explanation for why it is transcribed /ɪŋ/ in dictionaries is because it sounded more like [ɪ] than [i] in the accents spoken by the people who initially developed IPA transcriptions for English. The symbols used in phonemic transcriptions of widely spoken languages with a long history of IPA transcription such as English tend not to be updated unless there is a major reason they need to be updated, like an unpredictable split in the pronunciation of words.
Some previous posts about this topic on Stack Exchange:
Links to blog posts about this topic:
Previous writers on this topic seem to generally associate hearing /iŋ/ as a feature of the West Coast of the United States, but it is far from universal in this area.