How to reach websites that have failed while offline?
I don't think you are seeing anything out of the ordinary here. My connection goes down all the time, and it's never been immediate for me, but refreshing does bring the site up faster. 

My theory is that it just takes another amount of time for the browser to sort out that the connection is available again; the browser periodically pings the system to see if there is a connection available, and the lag you're seeing is waiting for the process to run its course again. Perhaps it pings once a minute as opposed to once every few seconds for fear of bogging down the processor.

Computers, as a rule, don't like it when things unexpectedly fail and thus take longer to get back up and running as opposed to when everything goes right.