Routing traffic when some of the pool members stop working

That is the intended behavior as the monitor takes about the 16-20s just to realize that the pool member has failed health-check.

You can explore the following options - SOL13310

Disabled or Forced Offline options provide you with the ability to retain existing connections and in the 2nd case, it removes all connections from the server.

You can utilize "disabled string" in the HTTP monitor to send information to the F5 that the pool member needs to be disabled - this makes the F5 realize that the member is going down immediately.

If you utilize OneConnect with 255.255.255.255 netmask, the VS will make the load balancing decision per-HTTP request rather than the default per-TCP connection and this will enable the F5 to make a load balancing decision for every incoming HTTP request and not just for the initial TCP connection.

I am not sure if I answered all your questions but a bit more detail on how you take a serverd down would be helpful. Also, it is essential to have a "process" within your team/organization that defines the method to take down a pool member :-)