What would you use to solve these problems?

VMs and load balancers?

From the documentation on the site it says that they're running on dedicated servers from Hetzner... So they aren't auto-scaling anything, they are paying for that hardware 24/7. It makes absolutely no difference what the number of running containers are, the cost remains constant.

You don't need anything. You choose the most convenient tool according to your professional judgment. I certainly hope that nobody is using Kubernetes because they are against the wall, and instead decide to use it for its features.