Is it possible to build a similar solution to managed MySQL cluster on a private cloud (VMware vSphere / Hyper V). Even with Kubernetes. A solution which supports auto-scaling, auto-upgrade, multi-master and similar to the largest public clouds offerings.
There shouldn't be any reason why you can't. I think that the issue would be in getting the memory management correct on the host. The orchestration would be the same.
Here's some resources:
https://www.digitalocean.com/community/tutorials/how-to-create-a-multi-node-mysql-cluster-on-ubuntu-18-04. It shows with docker as well. it shouldn't be hard to apply to vmware or k8s.
https://www.cockroachlabs.com/docs/stable/orchestrate-a-local-cluster-with-kubernetes.html to set up k8s locally.