Arduino ATmega328P works but board (Uno R3) seems broken
The R3 uses an ATmega16u2 in place of the FTDI. It is possible that it is not loaded, busted crystal, or it has no local power to it. It is a small SMT.

You could use your good Arduino setup as ICSP programmer to talk to the ICSP of the 16u2 to see it is alive. And if so then program it.