TT21 operation with no ICAO address

I'm moving this to a new thread since it has nothing to do with the
requirement for altimeter TSO.

I asked what happened if a mode S transponder was installed without an
ICAO address. That was a sort of tongue in cheek reaction to the idea
that someone would not want their tail number broadcast to the Feds.

Darryl responded:
A Mode S transponder absolutely has to transmit the aircraft ICAO
address, a correctly configured ICAO address is required for the
transponder to actually work--bad things might happen if two aircraft
had the same default ICAO address were being interrogated at the same
time.

I like to continue that discussion.

Isn't it true that a mode S capable transponder cannot respond to a
mode S interrogation if it has no valid ICAO address?

Isn't it also true that a mode S transponder is also required to
respond to both mode A and mode C interrogations.

If both are true then doesn't it follow that a mode S transponder such
as the Trig TT21, if installed with no ICAO address, will not respond
to mode S interrogations but will respond to all mode A and mode C
interrogations.

If that is indeed the case then any new transponder purchaser who
wanted mode S capability in the future, but was paranoid about
broadcasting the tail number, could disable mode S responses by
leaving the ICAO address entry blank and use it just like a mode C
transponder.

Andy