Tail rotor failure

In a way - it's called NOTAR. But it's used instead of the tail rotor, not as a backup for it.

Tail rotor failures simply aren't a frequent problem for helicopters. It's addressed by fixing the transmission's design, not duplicating it. If you go as far as to include major redundancies against its failure, you might as well use NOTAR or step up to contra-rotating coaxial.

Therac has answered the 'can you use exhaust to counter spin' part.

The reason why every helicopter does not have a tail rotor and a NOTAR ties back to a lot of issues with redundancy in aviation, that more safety systems can actually increase risk.

If we fit our tail rotor helicopter with a NOTAR system that normally sits at zero torque then now we need a system to detect tail rotor failure and start driving the NOTAR for Torque. This will kill people if:

• The system incorrectly detects tail rotor failure and activates (we get double counter torque).
• The system is designed to full disable original rotor when activated, meaning we now have a deliberately added weak link in the tail rotor drive making failure more likely (weak link failing or incorrectly activating).
• Tail rotor disable system fails in a way that jams the main rotor drive, making bad situation much worse.
• Making the tail rotor detection system safer and more reliable means it fails to act, or act fast enough to be useful (eg if issue was tail strike near ground).
• more interconnected hardware increases chance that maintenance/operator error causes an issue (and costs).
• The issue is actually the pilot controls, the pilot or the fly be wire system, so redundant controls not getting commands anyway.

All of the above systems can be mitigated by engineering in various ways but most of the failure modes cannot be removed entirely. Probably the most practical implementation would be a combined a tail rotor and a NOTAR where the flight control system takes pilot input as desired torque and feeds both tail systems to achieve that torque, if one fails then the other is driven harder. This still has constraints:

• The weight and thrust reduction from NOTAR is permanent, so engine or performance issues a lighter helicopter could have climbed out of will crash this hybrid.
• We still have failure potential where the 'desired torque' mixing system can fail in ways that leave helicopter uncontrolled.
• NOTAR and Tail rotor will have different 'feel' due the different response rates so still might not save aircraft due pilot confusion (imagine your steering wheel suddenly becoming half size on freeway - you could probably still drive but muscle memory would fight you a lot).
• Still does not work if original cause of issue was ground strike, mid air collision or main rotor boom strike, since you still need an intact tail structure for either system to work.

Possibly relevant reading at links in this Answer