I've written some posts on Kubernetes (K8s) as I think this will be an important technology in the next decade for database systems. Actually, it's not that I think Kubernetes is that important, but rather that the orchestration of containers that run database systems, along with the storage and networking, will be necessary. Kubernetes appears to be the most common and popular orchestrator, but that could change. There are other orchestrators out there.
Part of the reason that some technology gets adopted is that managers read about it and push it. Certainly technical staff have input, but managers have influence, and ultimately spend money on projects. While every manager wants to succeed and make their mark, no one wants to be left behind and have made a poor technology choice. As popular as containers and Kubernetes are, a little advocacy from management might get you moving in that direction. That's if you want to implement containers.
If your manager knows nothing about Kubernetes, can they still recommend the platform? Sure, but you can also help them to understand how it works and what the impact of this choice might be. I ran across A Manager's Guide to Kubernetes Adoption, which is a good read for a manager, perhaps tempering some of their enthusiasm to just install Kubernetes now. In fact, if you know nothing about containers and Kubernetes, you might give it a read as well.
I've changed my mind on database containers, and I do think Kubernetes has a future in the life of the data platform pro. It's not simple, and may not be easy to run for some time, but I think that will change. I do think we'll eventually just get a K8s service, or simple install, or use it in the cloud. I also think we may not need a lot more Kubernetes knowledge than is in the post, but we certainly will want to know what configuration items might affect our workloads, and certainly we'll still need to think about, and set limits (high and low) for our database systems.
I don't think that our world will change quickly, and certainly the more legacy applications you run, the longer your transition will be, but I do like the concept of Kubernetes as a platform where I deploy applications and services. I haven't quite felt that way about AzureStack to date, nor OpenStack and other variants, but Kubernetes seems to be a better idea for what I'd want. It will take time to get there, and it's not trivial effort to move to it, but once there, you should have reduced some of your infrastructure headaches.
As to whether your application will run better, look to your developers. All the hardware and well built platforms in the world won't work well if you don't write good code. Spend some effort there, and then think about moving to containers and orchestrators.