As a native Chinese, I can't produce 嚔 either. I'm interested in the statistics among college-educated Chinese how many _can actually_ write down that character...
I'm curious about that original passage - is the real reason for that something like that a native Chinese person writing a letter to another native Chinese person telling them that they were sick and had to cancel something would not phrase things like that, using the literal word for "to sneeze"? Is some other phrase or wording more commonly used to communicate things like that?
Dear Dr. Li, I'm sorry to say that I have to cancel our appointment this afternoon because I've been coughing and sneezing all day. I look forward to seeing you when I feel better again.
Modern Chinese would write 感冒 (gan mao; 'have a cold') instead. Much simpler to remember. Besides, they could send an audio message on WeChat, which functions like a global distributed walkie-talkie, or type it in pinyin, at which point the computer would auto-guess the characters and they'd only have to remember vaguely whether those characters "looked right" with some general degree of confidence, which is exactly how I input them myself. If they really had to write, they'd use the same method on their cellphone then copy the characters over. If the reader couldn't recognize the characters, an OCR system on their phone could do it for them. In short: through technology, this is now a solved problem.
I'd like to see the proportions of college-educated folks that could write it before and after the advent of the internet/commonly-used chinese keyboarding; were people better at it before the era of 'select the desired character from a list'?
Mainland Chinese use 嚏 instead of 嚔. I suspect they don't often know how to write that character because the phonetic root 疐 is only also used in 懥. Anyway, when writing, wouldn't Chinese use same grass-script style shorthand for it?