Salutations in letters

Whilst I appreciate that it is increasingly less common to write or receive a letter these days - and that traditional usage has been Dear Sir/Madam->Yours truly/faithfully or Dear Mr Smith ->Yours sincerely - the few letters rarely follow these “rules”.

I have had (1) Dear Mr Smith without any closure from the UK Pensions Service, (2) Dear Mr Smith->Yours sincerely from the local power board, and (3) Hi Mr Smith->Until next time from my bank. Personally I have never used ‘Yours faithfully’ (which smacks of subservience) since the turn of the century, even when applying for a job. I do still use “Sincerely” in a few emails (particularly when making a complaint). 

For the life of me, I cannot see why bygone formalities are still required for examinations such as the International English Language Test. 

As to emails, it seems more difficult to be formal. Mostly I use “Hi + first name” and end with “Cheers”. 

My question is what are  other people in English-speaking countries experiencing? Is stuff like “Yours faithfully” “Yours truly” now passé? If so is there any reason to teach them?