Except they are not extinguishing Linux as a whole, simply phasing it out of their own OS offerings. Linux itself is still alive and thriving, and will continue as such even after it has been replaced in Android and/or its successor.

The classic EEE strategy aims to eliminate a competitor entirely. If that was Google's plan (which I doubt), it was doomed to fail from the start.

Other replies have contested your use of the EEE metaphor, but I think it’s important to look at why Google seems to want to get off Linux. There are a few, but the biggest one seems to be the intentional lack of a stable driver API/ABI. That’s historically been a large part of the reason Android devices haven’t received software updates, since SoC manufacturers make their own “vendor trees” of the entire Linux kernel, and all kernel updates have to go through them. Recently with Project Treble they’ve tried to make things more modular; AFAIK it’s made it easier to apply small changes like security updates, but if you want to update Linux to a new major version, you’re still at the mercy of the vendor. Now, if you ask upstream, that’s the vendors’ own fault for not upstreaming all their drivers so that upstream can maintain them for them – but upstreaming drivers is hard, and it doesn’t mesh well with the (crappy) hardware development culture of proprietary and one-off designs, so they don’t do it.

In contrast, Windows and macOS both have stable ABIs for third-party drivers, with the result that you can upgrade your OS to the latest version and install the same drivers and they’ll probably work. One notable exception was the Windows XP to Vista transition, which made a number of breaking changes that required new drivers… and just look how that turned out.

If Microsoft switched Windows to the Linux kernel, well… To start off, of course, there‘d be a one-time break of all existing drivers; popular devices already have equivalents in or for mainline Linux, but there’s a very long “long tail” of devices that don’t. But even more importantly, they’d be opting into the Android model of periodic breakage for perpetuity. In theory, they might be able to avoid that problem by forking the kernel and adding their own stable APIs on top. But that would require such deep changes that the resulting fork would either (a) have an “abstraction layer” big and complex enough to be a kernel of its own, (b) be unable to keep up with upstream refactorings and thus diverge permanently and rapidly from mainline, negating the benefit of adopting Linux in the first place… or, most likely, both of those. Never mind the amount of engineering effort required to build that, compared to maintaining the kernel they have. Or the licensing issues they’d encounter in an ecosystem that’s currently even more proprietary-oriented than Android. Or the control they’d be giving up over future development directions (although to be fair, you could have said the same about Edge).

So yes, Microsoft could embrace Linux as their new kernel. But they’d be crazy if they did.

But that doesn't negatively impact all the prior uses of Linux. The "extinguish" part means that they ruin Linux for everyone else. If Google stops using Linux as the kernel for Android... so what?