So serverless meaning you don’t manage the db servers, so they may or may not put your stuff on the edge, cloudflare takes care of maybe load balancing too?

Yes, Cloudflare handles load balancing and neat tricks like measuring the latency between your users, your code and databases etc. and moving your code around the network to make it run fast: https://blog.cloudflare.com/announcing-workers-smart-placeme...

The vision of the Supercloud is that you give us your code and we'll figure out where and how to execute it: https://blog.cloudflare.com/welcome-to-the-supercloud-and-de...

IMO serverless is more about the pricing model more than whether a service is managed. Both Amazon RDS and Aurora are managed services, but Aurora is the one with the serverless pricing model.