Hi Mark,
Short answer: yes, the iphone will turn on with as little as a battery and a board.
Are you making sure that the device is powered off during the repair? Have you tried unplugging the battery and restarting the device?
You may have shorted something out during the repair. Here’s a quick test for you: unplug the battery, plug the new screen assembly in, plug the battery back in and hook the device up to a lightning cable to charge. If you do not get an image or the backlight does not turn on, wait a few minutes to see if you can hear the audible notification the device makes when plugged in to charge. If you hear the chime, the phone is booting up at least. Your new screen may be defective if this is the case, or there could be some board-level issues at play. Hope this helps!