I have no idea what it is like now but a significant part of the testing environment at Georgia Tech in the early '80s in the highly competitive engineering fields was "impossible" exams. These involved a timed in-class exam of an hour (sometimes two) where it was not possible to solve the major points problems analytically within the time limit, but a variety of approximation techniques were available. The highest scores came from understanding how to deploy the fastest approximation techniques.
Unlike many of my classmates I was really good at the mathematical part of the exam questions and could straightforwardly solve them analytically. However, this took too much time! I bombed several important exams learning that "good enough" is an important attitude to have when solving engineering problems. I hated this intensely at the time. (Never give that Stalinist institution a dime till I die.) BUT. There's a reason I went into numerical analysis in graduate school. Which I loved then, turned into a transient career that I enjoyed immensely, and remember fondly.
I doubt this approach is feasible for non-engineering disciplines.
Teach critical thinking before the college level - ideally well before. Much of the issues with overly relying on these tools boils down to a lack of critical thought - analyzing the output of these things critically, once you reach a certain competence of writing, will lead you to quickly realize you shouldn't overly rely on it, or maybe even use it at all.