How do you for sure tell that the logic board is dead?

There are basically two different approaches to a dead logic board, the modular one, based on parts replacements, suitable for the DiY amateurial level and the professional one, based on scientific approach to electronics, requiring electonics basic knowledge, tools, methods and so on.

In your case, as well as most others on this forum, all you can do is to try and buy parts, replace them, and see if the new ensamble works or not. It may be frustrating, or pleasantly surprising, but one thing one can be more or less sure of, is that nobody drops a 1000$ device that could be fixed with just a battery and screen replacement costing 50/100$.

Thus usually either the device has been already worked on and ruined by the owner, or has been handed to a repair shop without the skills to fix it (there are many around who “attempt” repairs). Unless you have the skills to diagnose the phone, you can change a couple of batteries to be sure you have a good one, put a new screen, replace the dock connector and hope for the best. If that doesn’t get the phone booting, the board is dead, at least on the DIY level, maybe a good technician might get it back to work but who knows, without testing it’s just speculation.