I don't think I've ever heard or seen it used by anyone except N. Korea.
Insofar as an archaic word is any word that evokes a sense of a prior time, and can be intentionally used to do so (e.g. "methinks"), then arguably yes — and that holds true even if usage increases somewhat after a popular reference, so long as the usage retains a degree of poetry/irony. That said, dictionaries mark words as archaic pretty conservatively, so I wouldn't necessarily expect to see it indicated as such soon.
The following interesting extract from the Grammarist tries to explain why “dotard”, a rare word, was brought back into light by a North Korean translator:
Dotard is a word that is rarely used in the English language.
The word dotard means someone who is old, weak and senile. The word dates back to the 1300s, derived from the verb dote, an even older word meaning suffering from senility, and the suffix -ard, which is an intensifier.
Today, the verb dote has taken on a different meaning, which is to be overly fond of someone.
Dotard became a word of interest in September of 2017 when North Korean leader Kim Jong Un used it in reference to American President Trump. Tensions had been on the rise between the North Korean government and the United States for some time over the testing and use of nuclear weapons in the Communist country of North Korea, and Jong Un issued a statement through the KCNA in response to Trump’s speech at the United Nations:
“Action is the best option in treating the dotard who, hard of hearing, is uttering only what he wants to say…I will surely and definitely tame the mentally deranged US dotard with fire.”
The word dotard is an interesting word choice made by Korean translator, as it is rarely used in English. Most people are more familiar with the word dotage, which is related to the word dotard and means the time in someone’s life when he is old, weak and perhaps senile. Someone is his dotage is not to be taken seriously, as his mental faculties are not as sharp as they once were.
I found this article by WP about 'dotard', in which they do talk about it being archaic.
But he may have pulled out an old dictionary.
(Meaning a current dictionary shouldn't include this word)
Here's their plot that shows that it fell out of use: