There is a solution : Use "failover" connection string : (not "aurora")
jdbc:mariadb:failover://<aurora-cluster-rw-endpoint>/db

When a failover occurs, that solution relies on AWS internal DNS to change to the newly elected master, but on automatic reconnection some operations will be transparently re-executed, like if the failed query was using readOnly, some not: an INSERT query that was in progress - driver cannot know if the command was successfully received by server or not.
(That will avoid creating a useless socket to replicas then. )
The only disadvantage, is that rely on the aurora internal DNS, connection will take a few more seconds compared to using "aurora" in connection string, but that's still a great solution.

Thanks for answering! This is documentation worth IMO as I'm sure the question will pop again.

Should I keep the finite lifetime of connections in my connection pool (as mentioned in the aurora section) or can I use the default infinite lifetime of the tomcat connection pool with the "failover" mode?

It's better to keep a max lifetime.

Technically that will work: tomcat pool will validate the connection each time it will be used, but if connection has been closed by server (after the time corresponding to @@wait_timeout variable), this validation will have to reach some timeout before automatic reconnection.

It's better to close connection silently after a time than to hang just when a connection is needed.

Good I'll keep this setting on then.

Once again thanks for your help, I'll close this issue and hopefully it'll come up in google if someone else search for it

Turns out I can't close my own issues so I'll let you reject it!

ok thanks, I'm closing issue then