"Yours faithfully" is just weird. Faithful to what? Why are you faithful to an unidentified individual or entity? Not judging you personally here; it's just weird. Sounds archaic, borderline creepy.
It's the traditional formal signoff where the recipient is not addressed. I understand it is of British origin. My guess as to its meaning is, since it is not addressed, you are faithful to the recipient that it will get to the right place. When I send correspondence externally, it is formal in nature. I have just followed templating conventions.
I think it is out of place for less formal communications.
"Cheers" to close colleagues.
"Kind regards" for other internal correspondence.
"Yours sincerely" for external correspondence where the recipient is named.
"Yours faithfully" for external correspondence not addressed to an identified individual.