Nice - Harrison obviously built some pretty good radiators and heater cores for them to last this long.
When I changed mine, I removed the heater core from the firewall side. Didn't remove the heater box from under the dash. Once the blower motor / outer cover was removed from the firewall, the heater core was easily accesible. However, when I did take it apart, the only seals on the heater core were the seal on the inlet / outlet pipe connections and what was left of the seal on the door. There was no obvious seal behind or in front of the core itself. From what you and John have pointed out, I got the seal on the wrong side of the core, but it sure made it fit nice and tight and is well cushioned. Only problem that created was that I had to manually seal up the gap with caulk.
One benefit of putting the pad on the dash side of the heater core was the alignment of the inlet / outlet pipes. With the pad on the dash side of the core they exited through the firewall properly. Without the pad on the dash side of the core, they didn't want to line up correctly with the cutouts in the outer cover - one of those repro issues - at least in my case.