I’ve thought this is an awkward and weird custom for as long as I can remember. I’m guessing it’s just a stubborn holdover from back when manners and formality were considered paramount. Something about letter writing brings that sentiment out in folks. But it feels very strange here in 2024, where informality and rudeness are the norm.
I speculate that it’s maybe serving two purposes. It’s a lazy convention that helps to “get the pen moving”. And it’s an acknowledgement that the person receiving the letter is probably dear to someone, even if they aren’t dear to the letter writer.
We’ve all seen those funny/awkward complaint letters that start angrily with “Dear sir or madam”. It reads as sarcastic-- and at the same time–old-fashioned and goofy.