Maybe that could be made to work, but if you're already committed to using a gas turbine of any sort, why not make it a turboshaft engine and build a conventional helicopter? Either way you have to pay for the expensive maintenance of a turbine engine. If a rocket-helicopter could have any real advantages (they're certainly cool, but I'm rather skeptical), it would surely be the simplicity of the mechanism making it cheaper to operate.
I think one of the big points is that elimination of the tail rotor could prevent 100% of the tail-rotor failure crashes. So something like 87% of crashes are related to tail rotor failures.
So I guess that would be the advantage. Also lighter because you're now missing an entire complex mechanism for the tail rotor and potentially that giant tail.
Coaxial rotor helicopters can be built without tail rotors, but my general impression is these are considered less reliable than conventional helicopters due to the added mechanical complexity of a coaxial system. But if tail rotor failure are that common, maybe coaxial rotor helicopters are worth the added complexity after all? They certainly seem more practical than rocket-helicopters.
I'm not sure about that; the Russian Ka-50 coaxial attack helicopter seems reasonably quick, certainly not the fastest helicopter ever but comparable to an Apache at least (315km/h vs 293 km/h respectively, though the Apache has a higher never-exceed speed: 350km/h vs 365km/h)
If you include compound helicopters (using horizontal pusher props or turbojets), coaxial helicopters can certainly be very quick; the S-97 can cruise at 410 km/h, but in that case you have the liabilities of a 'tail rotor' to contend with again. The S-69 could do a blistering 487km/h with it's turbojets, 289km/h without.
One to consider is the Eurocopter X3, a single rotor compound helicopter without a tail rotor. It could do 472km/h and used two tractor propellers with variable pitch to counteract the rotor torque.
Helicopters without tail rotors (Kamovs, Chinooks, etc) are typcially faster than conventional helicopters - They aren't loosing ~20% of their power to the tail.
