No. The radar transponder would have to squawk the assigned code or ATC would have known immediately the transponder was not reporting.
The other "black boxes" you refer to on this plane, do a myriad of things but most are passive collectors of data except the ELT.
The ELT would only be activated by crash or contact with water but not transmit in normal flight.
Question pfflier about the transponders. They can be turned off in the cockpit by the pilot, (or someone else), is what I do not understand. Why? Why have this transponder be even capable of being turned off? Why not have it located deep in the plane’s innards somewhere where it would continue it’s ‘work’ and not be available to in flight shenanigans ? Thanks in advance.