So the key fobs' buttons work to lock/unlock - this will be because a different module/antenna deals with this longer-range RF signal. It lets you start the car if you put the key in the designated slot - presumably so that the backup "old fashioned" transponder can be used (the ones normally in the ring around the ignition barrel of a car without smart keys) if the fob is completely dead. It also doesn't moan when you remove keys, and obviously doesn't respond to keyless commands.
Both fobs have the same issue, so it's not the fob. And all antennas seem to have the same symptoms, which in my mind rules out an antenna/wiring fault.
I am also going to assume the test that fails is relying on communication with the fobs.
Therefore, I wonder if the fobs have somehow disassociated themselves from the keyless system or vice versa? Does this diagnostic tool let you reprogram/resync the keys?