Can't reliably ping 6224 router from directly-attached system

Solution 1:

OK, I figured this out and I'll write it out here. This particular solution is unlikely to help anyone because it is an edge case.

Back in the ancient history of the link with this provider, we added a second VLAN to the primary one. At the time, the provider then connected this VLAN as both tagged and untagged on their side of the connection. Their switch treats the tagged and untagged as separate connections.

So what happens is my system connected to the Dell emits a arp broadcast (the management interface on this computer emits arp packets every half second for reasons which are stupid), which the switch forwards down the link to the remote site. The switch at the provider hears the broadcast on the untagged interface -- and sends it back to me on the tagged interface. The switch hears this and then concludes that the mac address originating the broadcast is really reachable via the provider's link. Follow up packets therefore get misdirected.

The solution was to have the provider change their configuration so that it agreed with that on the Dell. All general connection problems have ceased.