ADS-C is a two-way communications protocol; ATC can request a certain update interval and I believe also boundary conditions requiring extra location transmissions. If that communication fails, both the ATC and pilot are informed about the fact, and they can reliably switch to other means of communications (e.g. voice position reports over HF).
> These absolutely can be used for air-traffic control.
ADS-B is (at least for now) mostly an augmentation of primary and secondary surveillance radar, as well as something to give other aircraft better situational awareness (via ADS-B in), as I understand it. I don't think either ATC nor pilots are currently relying on it for safety-critical decisions.
The big problem is that it's always possible for some plane's ADS-B transmitter to fail (maliciously or otherwise). On land, you have secondary or primary radar to fall back to.
This applies doubly so to third-party (i.e. non-ATC/non-AOC) satellite-based relaying services, since not only the ADS-B transmitter can fail in that scenario (and ADS-B was not originally designed to be received by satellites), but the relay service could fail as well, and the failure mode would be much less visible to both parties than in ADS-C.
These absolutely can be used for air-traffic control. ADS-B has better temporal resolution than radar, and ADS-C is at least as good as manual position reports over the radio (used for procedural control over oceanic/non-radar airspace).
However, it's worth noting that these aren't really "transponders" (in aviation, that usually refers to beacon used for Mode-A/C secondary radar). Also, these are all active systems, dependent on the aircraft's electrical bus and easily disabled. In the case of MH370 the transponder _was_ powered down for some reason.
The only system that the pilot can't turn off is primary radar, but again, coverage for this system is limited once you leave populated areas.