Yes LXC and cgroups were Google contributions before the existence of Docker. But jails and containers in general were also general ideas before LXC and cgroups, I think most notably in BSD.
I feel like we must have had different experiences of that period 8 or so years ago. I distinctly remember the hype and bandwagon feeling of Docker's release. I remember various projects trying to capitalise on Docker to make PaaS-type projects; Swarm, Deis, Flynn (even my own Peas project), before Kubernetes came along. It seems so clear to me that Google were reactive rather than proactive. I really don't think Kubernetes would be a thing without Docker.
And even then I think you understimate googles contribution of "we've literally been doing container orchestration for longer than anyone else, here's tooling that is built with lessons we've learned in mind". That's far more than just "money and influence", and I don't think it's "exploitative".
I think in the same way that you're concerned I'm misrepresenting Google, I'm concerned that the documentary misrepresents the landscape of developers and projects that set the scene for Google's Kubernetes. Yes Google made significant contributions to the story that led up to Kubernetes, but crucially, I believe they didn't nurture the culture, they didn't create the mindset that makes developers (and managers) ready to think in a containerised way. And I think that's important because the smaller voices that did make those innovations easily get drowned out by the might of Google.
When kubernetes was released, the majority of that ecosystem didn't yet exist. Kubernetes influenced many of those best practices, standards, and tooling. So in shaping the ecosystem, yes, docker was catching up to Google. Like I think this would be a reasonable argument if Kubernetes was released today, but it was released only about a year after Docker was.
(heck, iirc, even docker as a tool was originally built on LXC, which I think google functionally upstreamed as part of borg hypervisor stuff, but I could be mistaken there. I do know that LXC was based on Google work though: https://www.nextplatform.com/2016/03/22/decade-container-con...)
Edit: Ok yeah I think I figured it out, cgroups is the borg upstreamed feature (https://news.ycombinator.com/item?id=9506800), LXC and evenutally docker are built atop cgroups, etc. etc. So the entire modern containerization infra is built atop a feature borg upstreamed.