Learning absolute pitch? It'something that's nigh on impossible. After many, many years as a muso, I can recognise, or sing, a single C note accurately 9 times out of 10. But that doesn't mean I have absolute (perfect) pitch ! Nothing like. Most folk are born with it - or not...
So the question is based on a questionable basis.
Fixed do means each and every note will have its own name, regardless of key. Do is always do, is always C. The other option is moveable do, where we take the tonic in a key to be do. Thus in key E♭, E♭ is do, in key F♯, F♯ is do.
The two can be very confusing - and if we add actual letter names into the equation, there are now three different ways to name notes! I work with a French band, and every muso there talks in 'fixed do'. For me, brought up on moveable do, it gets completely confusing, and could easily be remedied by everyone simply using the note names themselves. For example, a piece in key E♭ , talking about an 'F' note - in French that becomes 'fa', but to me it's 're'. Same note, same sound.
I think your premise is skewed, as even in fixed do, most folk are not able, or capable, of 'hearing note C'. If they were, they might have absolute pitch - and there wouldn't be any (pitch recognition) problems anyhow!
Of the two, moveable do has more advantages, as everything becomes relative, dependent on key, whereas fixed do by definition, must use 'accidentals' for some diatonic notes (ecept key C!), which creates problems when singing - single syllables suddenly become impossible for some notes.