I was the one who was told that it was rude, and that was in 1968 in the Deep South, so it isn't considered truly rude now -- it is more considered Too Much Information in terms of announcing that is where you are going.
Saying that you "need to use the restroom" is a bit like saying that you need to "spend a penny"; a polite little euphemism. In the US, women tend to prefer euphemisms that make it sound like we are visiting that room for some other purpose, such as powdering our noses or washing our hands.

(At my office, the usual expression is "run down the hall" because the toilet is located at the end of the hallway.)
The part that is considered somewhat impolite is making it EXACTLY clear what you are planning to do once you get there, and that is where mentioning the toilet comes in. As the poster I quoted mentioned, "toilet" here means the actual porcelain fixture, not the room that it is located in, so when you come out and say, "use the toilet" you are giving them a fairly specific mind picture. (Now if you said that you dropped your phone in the toilet, everyone in the room would be giving out advice about finding rice so that you could try to save it.)
Children tend not so much to say where they are going so much as they like to say what they need to do, because that gets an adult's attention more quickly, but they use little euphemisms as well, most notably saying that "I have to
GO" with very strong emphasis on GO. If you hear a child say this in an anguished tone of voice, he isn't just bored, LOL. "Go potty" is one of those childhood euphemisms, as is "go tinkle", and both of them are sometimes used by adult women who are trying to make a coy little joke. It will always draw laughter from bystanders if a child goes so far as to use more graphic terms in an audible tone of voice, such as "Mom! -- I gotta PEE!"
Oh, and BTW, that other four-letter P word with an "ed" tacked on the end does not mean drunk here -- it means angry. If you use the related expression in terms of teasing someone, expect blank stares; the expression "taking the **** out of" someone is essentially unknown here. (I'm dancing around on this one because the word is on the DIS blocked list, I think. In any case I don't want to get points while I find out.)