This broken portion of the PCI-e slot does not cover the x1 lane. It includes other important and necessary signals: +12V power, +3.3V power, SMBus pins, etc. See http://en.wikipedia.org/wiki/Pci-e#Pinout The PCI-e lane signals are after the notch (to the right in your picture).
You will really need to repair this portion of the slot to fix your card, for example by soldering individual wires from the broken gold fingers to a flexible PCIe extender.
That's a cool idea, but I'd be more inclined to see if I could attach something to replace the tab, then modify the riser cable to extend its pins upward, so that when you slid it into place, the pins would make contact with the board above the replacement tab. The replacement tab would only be there to provide something for the riser to hold on to. I'd really hesitate on soldering the board, but that's just me.