I've run production at a 'successful startup,' a million node corporate fleet, and a personal toy project or six, and consider Kubernetes the single most important work being done in the field right now. Docker, no (rkt is better architected). Kubernetes, yes[0].
Don't think of Kubernetes as a replacement for Puppet. Think of it as distributed init/supervisord with service discovery, which you are clumsily building with Puppet. Your automation ends up writing init, units, monit, supervisor, and so on, right? (You shouldn't be deploying apps with Puppet anyway because CD is not CM, but nothing like Kubernetes or Mesos really existed in the open source world until now so a whole generation learned to do it with Puppet.)
Kubernetes exists for when your scripted machine fails or you exceed the capacity of your scripted machine and need to horizontally scale. It is not a deployment tool per se; that's only part of what it does as a scheduler and resource manager. Every project, from personal to Fortune 500, needs to plan for those scenarios. I'm not saying Kubernetes is always the answer, I'm saying it solves a problem that every project universally has, despite your claims. The bonus of working that way is now you have an API to your machines and can treat them as a single unit of resources. Kubernetes is about half of building your own mini Heroku.
You've heard snowflakes versus cattle, right? Your way is for snowflakes. Kubernetes is for cattle. The threshold where one becomes more productive than the other is the constant debate, and as an SRE, my opinion bucks must people who talk about this on where that threshold lies. If you self-identify as a sysadmin you'll likely have one opinion, devops another, and then you have my group of crazies that generally want to crush all snowflakes. I can speak with experience that our crazy SRE ways scale pretty well to the hundreds of thousands of nodes case but also work quite well for toy systems.
Subscribe to everything CoreOS is doing. There's a spectrum of quality there, but they're running with the programmable infrastructure ideal and have largely bet the company on the Kubernetes ecosystem.
[0]: Kubernetes makes some interesting choices versus Borg and I think they're going to have trouble scaling it to that, and Borg really shines at Google because of the global filesystem layer that exists on every machine (they don't realize this and are doing gymnastics around storage in the open source side), but if they can mature Kubernetes it'll be a solid building block for platform development and eliminate a lot of the clumsy scripts and automation that you are defending and push us toward programmable infrastructure.