I would interpret 'side project' here as a work project that is not your primary and has low stakes for delivery into production timelines or expectations.
I don't really understand this. Work is usually prioritized. If it has low stakes for delivery into production then it should have a lower priority than other activities, but that doesn't make it a side project.
If you don't let developers prioritize some time to play around with new concepts and ideas and to learn how they work, they'll play around to learn in your products.
I feel like you're conflating multiple unrelated topics together. This isn't advice on how to use another team's experience, or cut costs, or maintain your team morale.
It is difficult to tell at a glance whether an engineer is qualified to effectively use a tool. Letting them self-train by working on self-projects in isolation compounds this effect.
The goal of this exercise is to give time and space to your devs to practice in a safe environment, while allowing them to push, deploy and review projects internally as if they were core products, so that other SMEs are allowed to spend some time every week reviewing those projects for smells and issues before those ideas make it into a core product.
It's true that running side projects on a company cluster in an environment where no one is quite sure how to use the tool properly is a disaster waiting to happen.
Fortunately, k8s can act as a very secure sandbox when it's configured properly, so you'll know how to mitigate such a disaster once your company has trained its engineers on how to use the tool effectively.
It's not mythical. Gmail and Google Calendar were both 20% projects. They were built to solve for bad internal web mail (mirapoint, I think... ?) and Oracle Calendar (which was absolute trash on Linux).
It's not as common as some would have you believe but it is real. A teammate spent his 20% on underwater topography on Google Earth and another spent his on the glider thing.
I'm not surprised 20% is not more common, but I am surprised among 20%-enabled companies that the norm isn't to have the company host side projects for all employees. Insurance, snacks, gym memberships, mobile plans and laptops, but not the one thing all hackers need?
Having an employee turn up a popular side projects while being vendor-locked onto your platform sounds like it should be more popular among rich people.
The way it is should actually be sold to the average entry-level fresh-out-of-college Google hire is probably closer to the way I framed it, though, than the examples of Gmail and Google Calendar which are pretty much two unicorns.
If you want your Devs to learn kubernetes you should pay them for doing it.
If you can't, hire a contractor with the expertise you need.