You're not saying the classicists confuse the smooth breathing with the acute accent, right? Only they can't figure out how to typeset them?

How do you enter them nowadays by the way? In the early days of the internet before unicode there were special fonts from SIL for example that first of all were using Latin characters (you'd type W and it would look like Ω) and secondly I think had the diacritics as separate characters, so the font would combine them. It was messy but at least you could type it on a normal keyboard. You could even read it in its ASCII form, most of the hacks were pretty reasonable.

Not smooth breathing and acute, but Unicode has two codepoints for acute accents (sort of). One is the modern Greek tonos, which often looks just like an acute accent but sometimes appears as a dot or straight vertical line, and the other codepoint is the ancient Greek oxia (the ancient term for the acute accent), which appears with all the other marks and their combinations (grave, circumflex, smooth/rough breathing, iota subscript).

Personally I type Greek using a vim keymap file, usually when writing LaTeX. I believe my keymap file is influenced by those older non-Unicode fonts, because I type w for ω and ;h for ή and >~h| for ᾖ. But my fellow students would mostly use Word. I don't know how they would type the letters, but commonly they would mix the tonos letters with everything else. I think what was happening is that Word would automatically substitute fonts that offered those codepoints, so it would end up showing Times for the letters with tonos and Palatino for the others (or something like that). Hence the ransom note effect.

My kingdom for a good monospaced font and supporting editor to handle these and RTL languages. I've found Setups that work with one or the other, but none seems to do it all just yet.

The combination of Adobe Source Code Pro & emacs works well for me.