’Bout or ’bout

I don’t know if anyone ever used this argument in favour of all lower case contractions I’m about to present:
the apostrophe omits the first letter of word (right, everybody knows that) and that it’s also a stylistic choice telling the reader that the letter replaced is silent or transformed to mimic colloquial speech, however in my view the apostrophe is also there as an replacement for that letter, so it should be lowercase. Here’s my logic:
Schematically: [replace ‘A’ with apostrophe]

  1. pro non-capitalization:
    replacement: About -> 'bout | reverse replacement: 'bout -> About :|: About=About [true]
  2. pro capitalization:
    replacement: About -> 'Bout | reverse replacement: 'Bout -> ABout :|: About=ABout [false]
    So as seen by virtue of logic and mathematical representation (rude attempt), how can #2 be a correct replacement?
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