I am kind of OK with this solution, and is in fact my plan to roll out HTTP/3 for my personal sites. I wrote https://github.com/jrockway/nodedns to update a DNS record to contain the IP addresses of all schedulable nodes in my cluster. I can then serve HTTP/3 on a well-known port and it is probable that many requests will reach me successfully. (I had to do this because my cloud provider's load balancer doesn't support UDP, and I don't have access to "floating IPs"; basically my node IPs change whenever the cluster topology needs to change.)

I don't really like it because it still means a minute of downtime when the topology does change. I would prefer telling the browser what strategy to use to try a new node, rather than relying on heuristics and defaults.