the reason that they say all of the solutions refer to the router is because its a network problem. The printer is going into sleep mode and the network isnt waking it up.
Now its possible the problem is with the printer. But for that to be the case means something is wrong with the wifi card or the firmware and your not going to fix that. The one thing that you can try on the printer is there might be a way to set it not to go into sleep mode. But that is going to waste electricity and maybe shorten the life of the printer.
Better to assume that its a network problem. The router is not sending or is blocking the command being sent from the computer to wake it out of sleep mode or is unable to find it on the network. or logs it off the network too quickly.
some things to check in the router are
A. if it has bonjour it should be set on.
B. multi cast should be set on.
C. the lease time for the IP address should be at least 24 hours.
D. lastly if you havent tried it yet. Just turning the router on and off can sometimes fix these types of problems.
I am available tomorrow to log in remotely and take a look at your router assuming it still has the default admin password ***** you know the password ***** it was changed.