I'm aware of this answer:
Today, "Your obedient servant" may sound extravagant and highly ornamental; but in the second half of the eighteenth century, when it first became popular, it must have sounded almost brusque, arriving as it did after many decades of truly elaborate declarations of loyal and grateful servitude.)
But I still don't understand why these words weren't regarded as disingenuous?
The Founding Fathers belonged to an elite social class, and weren't 'servants'. Why not write something politically correct, less classist like 'Acquaintance'?
'most humble/obedient' is too bombastic. The writer must know that everyone else closes with this adjectival phrase, and the writer logically can't be 'most obedient'; so now the writer redoubled his insincerity.
Examples. You can skip the following.
But I shall now wait your Lordship’s Arrival here, that if in any thing I can be of Service, I may be ready to obey your Lordship’s Commands; being, with the sincerest Respect and Attachment, My Lord, Your Lordship’s most obedient and most humble Servant
With great Respect, I have the Honour / to be, my Lord, your Lordships / most obedient and most / humble Servant
I have the honour to be with sentiments of the most perfect respect and esteem, your Lordship’s most obedient & most humble servt.,
With sentiments of the truest esteem & consideration I remain, Your Lordships Most Obedient Servan⟨t⟩