The /ɔ/ sounds (as in “caught”/“bought”) in RP and GA sound very distinct to me. The one in GA sounds more like /ɒ/ to me. Why isn’t it transcribed to /ɒ/ in the dictionary? And I wonder what the narrow transcription of “caught” in standard American English is.
Some American dialects merge the vowels of "cot" and "caught", however this is only marginally a geographic feature, instead it is generational x geographical. I don't make the distinction, yet relatives of mine (same geographical background) and random individuals whom I have asked do make the distinction. It is a much more common distinction in the Midwest, but even then in the course of 25 years, I observed a decrease in the frequency of the distinction in Ohio.
The choice of vowel in phonemic representations of this dialect is somewhat arbitrary, since it doesn't exactly match any of the performance standards for the IPA. It is extremely different from [ɒ], and quite different from [ɑ]. It is in between the vowels of caught and cot as pronounced by Midwest English speakers who make the difference. They are often transcribed as [a] and [ɔ]. Brooklyn speakers who make the distinction produce those two vowels very differently from suburban Chicago speakers, in other words, the actual pronunciation of the vowels cannot be represented with single letters. RP speakers have further differences.
Instead, if you insist on transcribing phonetic differences as opposed to measuring them, then you have to resort to those nudge diacritics that are available to indicate "less round" or "more round", "retracted" or "advanced" etc. You can combine those with elementary vowels such as [a ɑ ɔ ɒ].
I agree that at the phonetic level, [ɑ] would be a more appropriate base symbol (more round and less back). The vowel [ɔ] is found in "pork" and "walk" (="wok"). Then "pasta" is [pastə] except that the vowel is further back (it's not the same as Arabic "a" which is not [æ] and is not [ɑ]. Not everybody who uses IPA symbols is transcribing phonetically (also, not everybody consults the IPA-sponsored samples to calibrate their expectations). Compare how the experts pronounce [ɔ] and [ɒ] – performance-[ɔ] isn't very much like [ɔ] that exists in numerous natural languages.
The dialect that your dictionary refers to as "General American" has the "cot~caught merger", which means that the RP phonemes
/ɒ/ are no longer distinguished. As a result, "cot" and "caught" are pronounced the same. (The "father~bother merger" also merges in
As a result, most dictionaries will either use whichever phoneme is found in non-merging dialects (to make their dictionary more useful for other dialects), or apply one or the other consistently across the board. As a speaker with the merger, I tend to use
ɔ in broad transcriptions, simply because it's easier to type on my IPA keyboard. Or, if I'm also representing the father~bother merger, I'll use
ɑ, since it's the most peripheral of these vowels.
As to the exact pronunciation, it'll vary a lot, even within "General American"—a surprising amount of variation can fit under that label! Since there's no longer a distinction there, a certain speaker of a certain word might use something closer to
[ɔ], or something closer to
[ɒ], without it making a difference.
The THOUGHT vowels in RP and GA indeed sound distinct, but it's RP that has diverged from the cardinal [ɔ]. The THOUGHT vowel in modern RP (or Standard Southern British or whatever they call it) is close to cardinal [o].