I've always used the guideline that "business should always be faithful but friends are sincere." That, of course, fits generally with your assertion - if you know a person's name, you are more likely to be friendly whereas a "Dear Sir" needs to be "faithful."
My guess would be that "to whom it may concern" should therefore end "yours faithfully" because it is a busin
Generally speaking, I can't imagine starting a business letter with 'To whom it may concern'. My preference would be 'Dear Sir'.
I think you'd perhaps need to look at the purpose of the communication in order to decide how to end it. It might be that a simple name ,or even no name at all, might be appropriate.
Being American I can say that "Sincerely yours" and more commonly just "Sincerely" is AmE. I have never signed a letter with faithfully. In AmE faithfully is a salutation for lovers to declare their loyalty to one another. And I have written "To whom it may concern" hundreds of times, it is intended for those times when you have no idea who will be receiving the letter but you know that it
A letter to an unknown recipient needs no end salutation. "To whom this may concern" is not a form used much, except for open reference letters given to someone for use at any time.
Is it correct to use To whom it may concern as a start for a Letter of motivation in an exchange application? Or is the start Dear Madame or Sir more appropriate? And does it in that case have to end with Yours faithfully?