Do you know if the battery in the “used” phone was charged or not? It is possible that it is fully depleted. You could measure the voltage of the battery directly with a multimeter. A normal battery is ~3.8V, fully charged at 4.2V and the phone will not boot if it is below ~3V.
I’m also a little bit concerned about you mixing up the screws on the Screen bracket. Putting a long screw in the wrong hole can cause Long Screw Damage although in this case, your immediate symptoms don’t correspond.