Sorry about the confusion, by "completely dead" I meant "completely empty".

But now I am confused too. Let me see if I get it right. Your phone works fine unless the battery is fully discharged. When it is, it does not boot up. That is nothing surprising, you need to let it charge a bit (10 minutes should be OK) before you turn it on.

If that does not work, then it must be your USB port. Too much resistance either on the contacts or the solder pads results in too much voltage drop when the charging current is highest, such as when the battery is empty. The fix would be fixing the port, but an easier workaround would be never letting the battery discharge completely and/or invest in one of these.