The deep state is and always has been the ultra-wealthy.
Rampant, uncontrolled consolidation of media, PACs, lobbying, and granting cushy jobs to retired congresspeople all give the wealthiest individuals extremely outsized control over what happens in the US government.
I think the problem is that the "deep state" really came into public consciousness with Trump, on his first run. While I agree with your definition of the deep state, that is not what most people think of in current days, and Trump is probably the deepest of deep state you can legally be. He ran against the deep state, while being deeply embedded inside it. It was just easier to pass off because he wasn't a politician (at least from an American point of view, not sure of your country of origin).
It’s just funny he ran on releasing all of the Epstein Files, didn’t, and when we caught a glimpse of a slither of it, it turned out he was in there so much it could have been renamed the Trump Files
Rampant, uncontrolled consolidation of media, PACs, lobbying, and granting cushy jobs to retired congresspeople all give the wealthiest individuals extremely outsized control over what happens in the US government.