I don't think all those other estimates of Na were based on 'great experimental lengths' at least 2 of the numbers were derived from Kinetic Theory in the most part. Perrin's number was an experimental measure based on these theories and subsequent technologies such as X-ray crystallography have allowed far greater precision.
If measurement with greater precision shows your calculated value of 6.0222X10^23 to be inexact then I fear people are much more likely to go for the experimentally rather than the theoretically derived value.
I'm not sure you can say that there are a widely varying selection of numbers 'brandished around'. There certainly have been a wide variety of estimates historically but these aren't all still in common usage, people accept the need for a common value and the most reliable experimentally derived value is the one they go for, obviously rounded to whatever extent is appropriate.
I would contend with your number being wholly theoretical, surely the values you are calculating with from mass spectometry are experimentally derived. Since this value has a margin of error you may have to subsequently revise your calculation as a better value for the weight of the proton is reached. Essentially it is a question of whether you consider the mass spec or the X-ray crystallography to be a more exact experimental method. You have not shown anyway to derive the number which doesn't require considerable previous experimental effort.
If Na really was in line then that would indeed be a marvellous harmonisation with so many things we observe in nature, since it isn't however I think you would be doing a disservice to science, education and the many actual instances of this ratio in nature by trying to shoehorn Na in to try and make things pleasant, as Emerson said 'A foolish consistency is the hobgoblin of little minds, adored by little
statesmen and philosophers and divines'. I wouldn't accuse you of having a little mind but you seem to be tending towards the philosophical bent.
I think your point about expression on these boards is most peculiar. People are most welcome to say the most contrary and wrongheaded things if they wish, but they can hardly expect not to have percieved deficits in their thinking brought to their attention.
TTFN,
WK