I had a similar path as yours. Never had to work all that hard through school, did a degree at a university in the sciences which was somewhat the same (had to try a little harder after the reality settled in from freshman year...), but I had always just assumed that was the next step. This was just something I had to do to move on, so a high mark was the end result.
I was thrilled to graduate, took a job out in a semi-related field for about 5-6 years, then slowly began to get this feeling of "being dumbed down". Day by day, doing the same or similar things, not having to learn new stuff; it took a toll. I started listening to podcasts, and learning about all of the advances in science was enough to get me back into school and apply for a graduate degree.
Here is where I think this is something that people have to recognize they want for themselves. Starting my grad classes with other students, mostly fresh from undergrad, I could see a change in the value I took from the classes compared to undergrads. I saw my old mentality in them ("Ugh why do we have to learn this, it won't be useful, just have to get this out of the way to checkmark a box..."). Whereas I spent time in a career, and had to force myself to make a major life-altering decision. I wanted what they had to say, whether it was relevant or not, because these ideas could be weaved together in the future. The properties that dictate the science of one is not fully independent from the other.
This really became infectious. Everything I see, I try to learn beyond just memorization, but to a solid understanding and application level. "I'm running this test, the process isn't working, where is the most likely error? How can I optmize?"
Unfortunately, I don't think you can just teach this. It has to be something that someone actively decides they want to commit to.