i think so. that it could be push started means that the ignition/power/trinary switch, spark control, fuel delivery and ecu controls all work. I'm not too familiar with the immobilizer, but a look at the wiring diagram should tell you if the car can run while supposedly immobilized. maybe the starter is the last remaining suspect?
that being said, i wouldn't jump to replacing it before putting a multimeter on its poles while trying to crank in order to confirm wiring continuity.
I think you're right, the immobilizer appears to only kill the starter circuit. I've attached a copy of the colorized version of the 2005 Elise wiring diagram I made in case you don't have it already. Sheet 4 shows the immobilizer circuit, it sits between the starter switch and the starter solenoid (see on the right side of sheet 4 where the wires 194 and 188 go to sheet 1). It also ties into the multi-function relay unit at the fuel pump relay (see sheet 2) but I think this is only sensing, not interrupting. I hope this helps, enjoy the color wiring diagram.
2005 Ardent Red Elise, Gut Sport wing & splitter, custom diffuser, Sector111: DSBrace, gPan3, subStuffy2, Larini Sport muffler, decat pipe, Schroth Racing 5 pts, wheel studs, Team Dynamics 1.2 with Hoosier A6s, Koldflash 200 ECU flash, BOE rear clam hinge, BC Racing single adjustables, Hawk pads, SS Brake Lines, HID lights, custom tail and high mount brake light...and much more...