Hm, I now recall a similar issue, not with a router, but with a switch.
(Basically, a router is a switch with some added functionality, like a firewall...)
The switch had a bad capacitor, which only failed if more than one computer was connected to it.
With just one computer connected to it, it would do fine.
I replaced the capacitor, and all was good...
So, in the end, yes, the router is the main suspect here...