Additional SQL Server features and topics not covered by specific categories
The problem went away as mysteriously as it arrived.
I tried all kinds of things, yesterday. I checked everything I could think to check. I pinged and telneted. I systematically stopped and started different parts and services while carefully checking the results. None of that showed anything unexpected or particularly interesting.
This morning, I fired up A and had no internet. I rebooted and had internet, but the keyboard wasn't working. Now, while I've had A for a few years, the keyboard has had an issue or two over the years, so I had seen that at least one time in the past, so I rebooted again and got both internet, keyboard....and SSMS connected to S quickly and without issue.
So, the problem has vanished, and thinking back on it, the only thing that fits is that rebooting A solved the problem, but why was the problem there? It could be A, it could be the router (which is new), or it could be some interaction between the two. At this point, I don't have enough data to be certain, nor is there anything that needs to be fixed, at this time, but it's still a mystery.