I'm thirty something. How did my teachers engage me in doing math? How did they engage me in rote-memorizing the multiplication tables when portable calculators were already a thing, being operated by coin-cells or little solar panels?

Part of teaching is getting kids to learn why and how things are done, even if they can be done better/faster/cheaper with new technology or large scale industrial facilities. It's not easy, but I think it's the most important part of education: getting kids to understand the subjacent abstract ideas behind what they're doing, and learning that there's value in that understanding. Don't really want to dichotomize, but every other way kids will just become non-curious users of magic black boxes (with black boxes being computers, societal systems, buildings, infrastructure, supply chains, etc).

The same way you did so before LLMs existed - you rely on in-class assignments, or take-home assignments that can't be gamed.

Giving out purely take-home writing assignments with no in-class component (in an age where LLMs exist), is akin to giving out math assignments without a requirement to show your work (in an age where calculators exist).

Many years before LLMs were ever a thing, I recall being required to complete (and turn in) a lot of our research and outlining in class. A plain "go home and write about X topic" was not that common, out of fear of plagiarism.

The same way you did it "in the age of the internet" and "in the age of TV"

You largely don't. Of course you try to motivate them. And it does work on some, mostly out of fear of an authoritarian institution that's setting the course for their life, rather than conveying the intrinsic value of the pursuit. And the rest is just glorified day-care.

Sure, use AI for research, just like using the Internet for research.

But don't copy/paste AI generated content in the same way that you don't copy/paste a chapter from a book and pass it off as your own.

Invert the assignment, provide a prompt to supply to an essay writing AI of the students choice, but the assignment is to provide critique for the veracity and effectiveness of the generated essay