It does work without the battery. I had no problem plugging it in and testing it without the battery connected.
I just finished doing this repair yesterday and I have a few tips for people trying this repair. First of all, when removing the display, be really careful on the sides of the camera assembly, because this is where the antennas are located. These antennas are sandwiched between the adhesive that holds the display on and the frame of the device, so it is very easy to break them when you are opening up the device. I actually did end up breaking one of the antenna “flex cable” style pcb’s, but luckily I didnt break any of the traces and my antennas seem to work fine. Also, test the antennas, and make sure the device is able to charge at least a few percent before gluing the display back in place, as the battery might not be able to charge if the connectors to the battery aren’t all connected. This is a very difficult repair, and it feels like Microsoft tried to make the battery one of the most difficult parts to replace. Getting the display off is hard, but it is only the first step.