'Small' Databases are the first, and are easy to dump into kubernetes. Anything DB with a total storage requirement 100GB or less (if I lick my finger and try to measure the wind), really, can be easily containerized, dumped into kubernetes and you will be a happy camper because it makes prod / dev testing easy, and you don't really need to think too much here.
'Large' database are too big to seriously put into a container. You will run into storage and networking limits for cloud providers. Good luck transferring all that data off bare metal! Your tables will more than likely need to be sharded to even start thinking about gaining any benefit from kubernetes. From my own rubric, my team runs a "large" Mysql database with large sets of archived data that uses more storage that managed cloud SQL solutions can provide. It would take us months to re-design to take advantage of the Mysql Clustering mechanisms, along with following the learning curve that comes with it.
'Massive' databases need to be planned and designed from "the ground up" to live in multiple regions, and leverage respective clustering technologies. Your tables are sharded, replicated and backed up, and you are running in different DCs attempting to serve edge traffic. Kubernetes wins here as well, but, as the OP suggests, not without high effort. K8S give you the scaling and operational interface to manage hundreds of database nodes.
It seems weird to me that the Vitess and OP belabour their Monitoring, Pooling, and Backup story, when I think the #1 reason you reach for an orchestrator in these problem spaces is scaling.
All that being said, my main point here is that orchestration technologies are tools, and picking the right one is hard , but can be important :) Databases can go into k8s! Make it easy on yourself and choose the right databases to put there