Fish are not just NH3 factories, their wastes contain all the breakdown products of their food which they did not or could not absorb for their own metabolism, these can be sulfur compounds and organics, neither or which we can measure easily or cheaply. They secrete hormones which while in nature would be flushed out simply remain in the water and build up. None of these are good for the fish but sadly we cannot measure them. But all is not lost. We can measure the NO3 that is produced from the NH3 that they secrete and based on this, we can make a hand-wavey generalisation about the other pollutants in the tank. To my knowledge, no one has actually measured the precise relationship between meausured NO3 from fish secreted NH3 and concentration of fish derived pollutants, but it doesn't matter, we can choose a benchmark NO3 level, like 20ppm and adjust our maintenance and water changes accordingly.
So NO3 is not just the non-toxic (in the short term) end product of 'cycling', it also allows us to have some idea of the level of pollutants in the tank.