> Now I think I kind of have an answer. Maybe the level of diagnostic refers to the LLM effort level....
The real answer is that it was gibberish. IIRC, the writers would insert "[tech]" in the scripts, to be later fleshed out into technobabble.
What you say is ret-con at best. A more realistic contemporary interpretation of the "level 3 diagnostic" technobabble, would have been a particular pre-programmed diagnostic procedure (like "run the POST memory checks" on a PC).
I thought the ship's systems were just instrumented with self-diagnostic hardware and the "levels" of diagnostic indicated varying levels of time/effort.
Consider modern hard disks and SSDs, which are fitted with SMART self-diagnostic tools. You can run a short test, which checks the various electrical and mechanical (for HDDs) components, environmental sensors, etc. and ensures those are operational at a high level. Or you can run a long test, which checks the platter surface/flash memory sector by sector and can take several hours to complete, but which will flag up more problems earlier. Presumably a warp nacelle, being a much more complex and dangerous system, might have many more diagnostic procedures that can take varying lengths of time, from basic integrity checks of the plasma conduits to the warp core to ensuring the power flux coming from each coil is within tolerances.
Obviously, yeah, it's fiction and it's just bullshit. But one of the nice things about TNG is that its premise pretty much confines actually intelligent machines to Data and his relatives, with a handful of exceptions such as Moriarty, which the Holodeck generated specifically to challenge Data. Thus, it's designed around the idea that there's always something for the biological crew to do. Hence, "level 3 diagnostics" or any diagnostic are automated data gathering operations similar to SMART tests, allowing the engineering crew to perform analysis on what the data mean. Star Trek is fundamentally humanist; its fictional universe is constructed to celebrate the greatness and achievements of humans and other living beings, without going to the extremes that Dune went to by banning computers altogether.
Now I think I kind of have an answer. Maybe the level of diagnostic refers to the LLM effort level. Great. So then what are all the people on the Enterprise actually doing? My guess is they're all just being people and doing people stuff like exploring the galaxy and smugfacing the Romulans. It would be easier to just create a race of Datas to do it safer. The Datas could probably be Bluetoothed to the Ship's Computer for even more optimization.
I think some of the struggle is that as we summit each subsequent hill of challenge, the sunset (or is it a sunrise?) is becoming inescapably bright. We're faced with that big ol' "what's the point of all this?" question. What's the point of studying math? What's the point of exploring the galaxy? What's the point of standing under a stack of blue barrels?
I'm not a math guy. I dunno if a Fields medal comes with free Taco Bell for life or something else that would truly move me. But I love that I get to read books and learn cool things and feel that spark of joy when my understanding expands, even if the ending is right there with nothing stopping me from skipping to it.