From time to time, I encounter historical letters that close with a long valediction, usually a variant of "I am ... your servant". Thomas Jefferson's valedictions are very noticeable in their praise.
When did this fall out of favor? I queried the ngram viewer to compare the phrases "humble and obedient servant" alongside contractions of these phrases such as "yours truly" and "yours sincerely". I can observe a rise of these contractions during the mid-19th century along with an eventual decline until the present day.
What are some possible reasons why have English speakers removed themselves from these styles of valedictions?
The valedictions in question never truly fell out of use, but have become abbreviated instead.
Valedictions in English writing have traditionally been very formal affairs following the French model. You can still observe similarities between 18th c. Georgian English valedictions and their modern counterparts in French:
My Lord, Your lordship's most humble, most obedient servant, Sam. Johnson, "Letter to Lord Chesterfield," Samuel Johnson, 1755
Nous vous prions d'agréer, Monsieur, nos salutations distinguées, Modern French, Calliope Source
English has long had a tradition of abbreviations although valedictions, as a piece of formal writing, withstood most of them until the 19th century. These long valedictions were eventually whittled down to the common form we see today:
There remain a few, rare times when the full English valediction is still employed. Diplomatic correspondence tends to be very formal indeed and it was considered good manners to conclude letters to ambassadors and Governors-General in Commonwealth realms with the full valediction until the early 1980s. As of 2000, Debrett's still mentions the full valediction although it described it as somewhat "old-fashioned" even then, so it may be totally out of use today although I wouldn't be surprised if it was still used sparsely in official documents between the Foreign Office and accredited ambassadors to the United Kingdom.