Why is it not "Sorry" instead of "Excuse me" after somebody sneezes or does something that is improper to people around?

I've heard people saying "Excuse me" after sneezing in public, but should it be "[I am] sorry" instead?

An answer to Difference between “Excuse me” and “Sorry” says that:

"Excuse me" is asking for permission. "Sorry" is asking for forgiveness.

So, if you want somebody's pen, you will ask like:

Excuse me, can I borrow your pen?

And, if you found a pen but you didn't find its owner and, due to your emergency, you used it and later explained it to its owner like:

Sorry, I had to use your pen in your absence. It was an emergency, sorry again.

So, following the above concept, saying Excuse me after sneezing will be like doing something first and asking permission for it later. What is wrong with saying Sorry here instead?

Solution 1:

See this definition from Webster.com

5 —used as a polite apology for a minor fault or offense, such as laughing, coughing, or burping
6 US —used as a polite apology for getting in someone's way or bumping into someone Oh, excuse me. I didn't notice you standing there.

I think the answer you quoted in the other question is not really correct. "Excuse me" is not generally used to ask for permission. The example you give is of one of the following definitions:

1 —used as a polite way of starting to say something Excuse me, but do you mind if I shut the window?
2 —used as a polite way of starting to interrupt someone Excuse me, but may I say something?

They're basically introducing the request by apologizing for the interruption.