Why would my Kenmore laundry center have black/yellow oil/grease sprayed across the inside walls?
The only thing in a washer that has significant amounts of oil is the transmission. At this point, if the machine seems to be running normally, then let it be. That is what the appliance repair guy was talking about.

If the transmission dies later, the worst that can happen is you burn up another belt. So unless you want to spend the money on a new transmission now, just wait until it fails.