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 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.
From Benjamin Franklin to the Earl of Loudoun, 3 March 1757
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
B Franklin
From John Adams to the Marquis of Carmarthen, 28 May 1785
With great Respect, I have the Honour / to be, my Lord, your Lordships / most obedient and most / humble Servant
John Adams.—
From Thomas Jefferson to Lord Wycombe, 25 July 1789
I have the honour to be with sentiments of the most perfect respect and esteem, your Lordship’s most obedient & most humble servt.,
Th: Jefferson
From George Washington to the Earl of Buchan, 1 May 1792
With sentiments of the truest esteem & consideration I remain, Your Lordships Most Obedient Servan⟨t⟩
Go: Washington
That term originates from the Latin 'humble servant' greeting servus humillimus, which was used all over Europe from the Middle Ages forward. By the 18th century most people using it, in whichever form, weren't servants.
It was prevalent to the extent that 'servus' is a common greeting today in Czech and German (particularily Austria) and in the form 'szervusz' is in fact the most common greeting in Hungarian. The translated version 'tjenare' is a common casual greeting in Swedish, a shortening of "eder ödmjuke tjänare" which was how they ended letters in the 18th century. Just as the French would sign off with "votre humble serviteur" or even "votre très humble et très obéissant serviteur", or Germans with "dein demütiger Diener", "ganz gehorsamster Diener" or "ergebensten Diener".
So no, it wouldn't and couldn't be considered bombastic or disingenuous to use the same valediction that much of contemporary England and Europe was using. You're interpreting it more literally than they would have because you're not as accustomed to seeing it everywhere as they were.
As Platypuskeeper notes, you don't see this very often so it seems excessive. There was a recognition that it took far too long to write, which is why you commonly see it abbreviated , e.g. " yr h'mble & ob'd'nt &c."
And of course George Washington was not, in fact, offering to be your servant, anymore than a French bureaucrat rejecting your application actually is saying that the deep emotional bond between you causes her real pain when she signs off with " avec ma sympathie entière"....just like in English we assume the same with "we are very sorry to inform you".