AWS Aurora RDS 2nd reader gets no queries
We have also faced the same issue and it got resolve after reduced the DNS TTL value in our Java application.

Amazon Aurora for MySQL or PostgreSQL distribute workload between reader nodes as below (From AWS Manual)


  Aurora uses a  DNS based round-robin load balancing mechanism for
  balancing connections between the reader instances.  This load
  balancing is handled through Amazon Route 53's DNS service.  This
  means that your application makes a DNS query for each new connection
  request. Each time the reader endpoint is resolved, the instance IP of
  one of the reader is provided for the connection. In order to get the
  instance IP of a different reader each time, one must keep resolving
  the endpoint without caching DNS. Aurora DNS cache uses a short Time
  to Live (TTL) of  5 seconds. To ensure both the readers are equally
  utilized, care must be taken not to increase the TTL more than 5
  seconds for a client and network configuration. Caching DNS responses
  for longer than 5 seconds can result in overutilization of one reader
  over others as you have been experiencing.


In your application check the DNS TTL set for the application and adjust it down to 5 seconds.