It's often taught in political classes and it isn't a conspiracy that there exists a "deep state" or "shadow goverment".
For example, the roman elite troop, the Praetorian Guards, can be classified as a "deep state", as they had their own goals and if they were not meet, then they would just kill the emperor. At the same time, the emporer and the senat had almost no control over the Praetorian Guard.
There's nothing wrong with the phrase "deep state". If you're concerned about associations with Trump (which I think you should not be), you can use a former Obama official's favored term, "the blob". Both, and many other similar phrases, are common in the foreign policy literature.
I think the first time I heard the phrase "shadow government" was in connection to Iran-Contra, where it described Oliver North and the small group in charge or those operations, but it's a phrase with many uses. I can't remember when I first heard of the "deep state", but I'd be fairly certain it was before Trump or anyone connected to him started using it.
"The Government" isn't really a single institution, but a surprisingly loose collection of various institutions that all have some level of momentum guiding the direction they move in. It's critical to note that these institutions may have interests that harshly conflict with one another. Even the intelligence community, insular as it is, cannot be thought of as a monolith, but as various institutions with various priorities, some which conflict with the interests of other government institutions.
When the government does something we dislike politically, or does something we don't understand, it's "the deep state" or "the shadow government" which betrays a deep misunderstanding of what the government is. There's no central, shadowy cabal controlling the affairs of "the government" from behind the scenes, the various institutions that comprise the government maintain momentum and do what is in furtherance of their particular interests.
J. Edgar Hoover was a perfect example of what people are talking about with the deep state. You can claim that’s just “institutional inertia”, but it’s still some unelected official with massive power wielding it in nefarious ways.
It's a very imprecise term covering bureaucratic norms and blindspots, to unsanctioned domestic and international terrorism, so it's not really useful except as shorthand with ideological bedfellows, so-to-speak, and you're right that it's often used to mean "the government guys I don't like but am too lazy/ill-informed to describe". But OP mentioned Gladio -- I think that's context enough for this thread. Do some reading on the topic (beyond Wikipedia) if you haven't already, and you might get a sense of what this non-centralized deep state can do.
And I didn't make it clear, but I brought up the North et al "shadow government" as an example far on the small-scale end of the spectrum of meaning of "deep state/shadow government". It _was_ a centralized shadowy cabal doing a now-documented conspiracy. People argue about how much authority they "seized" and how much was delegated by Reagan (and how much capacity he had to delegate much of anything), but regardless, it was not democratic to illegal go around Congress. What I've described as the "deep state" is the more common usage, and it's on the "amorphous, non-centralized, competing institutions, largely free of any democratic accountability" end.
To me, 'deep state' implies an illegal conspiracy against democracy. 'The blob' is a large organization that is hard to change, which describes every such organization. Ask an executive at a Fortune 500 company about blobs. The executive branch has the added complication that the CEO has limited powers and doesn't make the rules, Congress does. At work, you follow the boss's instructions. In the executive branch, you follow the Constitution first, then the laws made by the American people via Congress (accumulated over centuries), then the President's instructions - including instructions of prior presidents that haven't been changed.
Maybe 'deep state' was used before Trump, but the meaning has changed.
This is a very new, post-Trump/QAnon interpretation. The “deep state” has, for at least many decades prior, meant those aspects of government which persist from administration to administration, largely unaffected by those in power. E.g. the CIA keeps doing its thing regardless of whether a Democrat or Republican is in office.
Now in as much as these aspects of government are controlled by unelected officials who are willing to work against the elected representatives (think: J Edgar Hoover), that constitutes a shadow government.