Replacing front panel. Pressure contact came off of logic board.

Unfortunately there is no easy answer. The loose clip or EMI finger, as you mentioned, is intended to make a conductive contact between the logic board and the metal outer frame. I can't say for certain how critical this contact is, but I expect it is intended to reduce interference with the phone's radio signals. You might get away with a functioning phone with out the tab, but it may experience more connection troubles.

Assembly and disassembly of the iPhone 4 is a bear (as you know by now) so a reassembly just to try it out without the little tab may not make sense.

It is hard to tell from your photo of the back of the logic board what has been left behind where the EMI finger tore off. If the circuit trace is still there you may be able to resolder the component with a very fine tip iron. If the circuit trace was peeled off with the EMI finger the repair will be more tricky and require that you find some way to expose a fresh bit of the circuit trace.

Electric conductive adhesive could work but the good stuff can be difficult to find retail, it isn't cheap, and often isn't in tiny quantities. Master Bond produces all sorts of adhesives with a variety of properties. To find it or similar products you will have to find an electronics shop that is more savvy than Radio Shack, and with luck such a shop might be able to help you chose an appropriate adhesive. I would avoid the cheapie stuff like wire glue and other things sold as liquid solder. Most of these will make a weak mechanical bond and some introduce too much resistance.

Unfortunately finding a replacement iPhone 4 logic board will be difficult or expensive. With all these road blocks, trying it out without the EMI finger looks more attractive.