I've been taught to write "Yours" ever since I started writing letters. But today I realised that "your" is an adjective qualifying the person who is writing the letter. Now, since the adjective for you is your, why is yours always used?
If the valediction ends with a noun, then the s is omitted, as in:
Your obedient and humble servant,
If the phrase uses an adverb, then yours is used:
There's nothing tricky about this; just think about how you'd say the full sentence, if it began with
I am your closest ally.
I am truly yours.
In the 18th and 19th centuries, many letters ended with the full sentence. More here.
I could understand what your doubt is here. 'Your faithfully applicant' is right if you would like to put here YOUR as a possessive adjective. "Yours faithfully" is an adverbial phrase. It is not a sentence, just a salutation from an applicant. The expansion of this phrase is "I am yours, being here faithfully" It's standardised phrase. like "I am afraid".