I have been surprised at the acceptance of the phrase "deep state" (regardless of your opinion of the topic) because it simply sounds like a foreign phrase to the English-speaking ear.

In the US, and in English in general, the use of "State" to talk about government is largely confined to political scientists and the odd reader of Leviathan. In popular culture, terms like "bureaucrat" are more typical, oftne with "unelected" tacked on the front. And of course in the US and Australia, at least, "state" is generally used for major subnational government divisions ("state of Texas").

On the other hand, the popular use of "state" as an actor or participant in its own right comes from 19th century revolutionary theory, especially communism. So countries heavily influenced by those movements (russia and former Warsaw-pact bloc, China, even Turkey, et al) use this construct naturally.

So it's fascinating that this loan-word has become popular in English.