Universal advice doesn’t apply to everyone. Your pet peeves aren’t someone else’s.
As someone who’s been an entrepreneur for almost 2 decades, I’ve used these phrases without issue.
I see no problem with starting out with an introduction. The difference is: Are you spamming, ergo contacting someone who never signed up to be contacted by you? Or, are you contacting someone who’s already opted-in? If it’s the former, then you should absolutely introduce yourself. It’s a professional courtesy, in my opinion. Who the hell are you, and why are you contacting me? If it’s the latter, then of course, no intro necessary.
As far as a polite intro such as “I hope this note finds you well”: This became a very common intro especially during a pandemic when we are concerned about others — or at least we were at one point. Personally, I would rather someone send me a note that starts with that, than just going for the throat with their selfish intentions, e.g. to sell me something.
Telling someone you hope they had a great holiday shows you actually care about their lives, and again that you’re not simply coming in hot just to sell them something.
As someone who had a PR agency for almost 20 years, I never had any issue with a polite follow-up note. People get busy. Emails get lost. Don’t let the pet peeve of chosen words overshadow the intent.
And if you admit that you are imposing on my time, I appreciate that.
Also, personally, I would like to see suggestions on what people could do instead. Call me old school, and maybe it is a generational thing, but primary over the phrases you mentioned: abrupt, impersonal (or overly chummy ones), and text-like emails — especially if I don’t know you from Adam- will always end up in my deleted items folder.