I suspect the answer is lurking in the question, but the addition of the word 'quite' is clouding the issue.
Try 'John's a hero' alongside 'John's the hero'. The first, 'a hero' carries an implication that the quality of heroism in John is an assessment or judgement made and held by the person speaking. Whereas, the second version 'the hero' carries an implication that this is a title of acclaim, given and held to be true by the speaker AND others, or perhaps only by others (when said with a certain dismissive inflection), or perhaps by nobody (when said sarcastically).
Essentially it's a subtle shift in meaning from a personally held judgement of quality which doesn't carry so much weight because it remains subjective, to a recognition of a shared (and usually more emphatic and influential) judgement of many. The dismissive and ironic versions follow naturally.
In a dismissive sense, the speaker might even do that gesture with the fingers in the air to indicate quote marks around the words 'the hero'. But in the first two senses ('a hero', and 'the hero') there is no special emphasis in speech.
Perhaps in American idiom the distinction is clearer. One might say, "John's a man" which might be stating the obvious, or a contraction of "John's a (grown-up) man". But it is obviously different and less emphatic than saying "John is -the man-", or "He's de man" in vernacular. The latter clearly suggest a public profile and acclaim.
The use of 'quite' is, I suspect, more of a British thing these days, but very few would recognize it is the same word as 'quit', as is 'quitted', 'quieted' or 'acquitted' which is essentially to say 'resolved', 'settled' or 'accounted for'. The original Latin for 'quit' suggesting 'quietening' or 'settling' was applied equally to natural events (which were attributed to relations between the Gods, or between the Gods and man) and to human relations and obligations and debts. Eventually to 'quit something' was to settle or state the account of something, or keep a 'good tab' (a credit standing or rating). So to 'acquit yourself well' was to be in 'good standing', and for something to be 'quite' was to suggest that the accounts had been audited and whatever was 'quite ....' was something that had been calculated and accounted.
In saying 'quite' the speaker lends a subtle level of abstraction to the statement or value that follows. To say 'he is quite a hero' is essentially to say 'he is a hero by my calculation'. And 'he is quite the hero' comes out as, 'he is the hero by popular account(ing)'. Rather than suggesting that 'calculation or accounting' makes it even more irrefutable and certain (as we'd expect), there's a very faint suggestion of a modest or muted result. The assessment is not drawn from the passion or emotive effect on the speaker or public, but derived from the somewhat dry 'book of accounts of human actions and intentions'. It is a way that British speakers might keep emotions at bay in their speech, as in:
"Carruthers, our chaps have just climbed Everest backwards, what do you say to that?!"
"Well yes old bean, quite."
In the case of a parent speaking of their child, the addition of 'quite' allows the preservation (or simulation of) some kind of objectivity and (perhaps false) modesty. They are saying, "It is not because I say it is so, or because I feel it is so, but because it is accounted (and audited) and this is the result that has been tallied up." All of this dissimulation is largely alien to American English I suspect, and certainly to Australian English.