In principle, you could use ammonia, but there are practical disadvantages.
The sulfur-based odourants are some of the smelliest compounds known and are detectable in very low concentrations. In practice, they are used in concentrations of perhaps 30ppm. Several percents of ammonia would be thousands of times more than this concentration implying greater expense. At this level, the concentration of ammonia would be problematic because of the corrosive nature of certain types of pipe and would probably create other safety concerns (it might even be a direct safety hazard at the concentrations where it is readily detectable by smell).
Typical sulfur-based odourants are detectable at such low concentrations that they are not a direct safety hazard. Nor is there much worry about the combustion products at such low concentrations.
Ammonia is also much more reactive than the typical sulfur-based odourants and would be much more likely to be stripped from the gas during transmission which defeats the purpose.