The idea behind the Ilford method of short fixing times with FB papers is to prevent the fixer from entering into the fiber base. Therefore the slow removal from the paper is avoided. Ammonium thhiosulfate removes silver halide faster than does the sodium salt. It also washes out faster than sodium salt. This allows the Ilford method to use a higher concentration of ammonium thiosulfate than is normally used.
As you probably know the theory behind a two bath fixing system is that the amount of silver in the second bath never reaches the concentration that exists in a single bath fixer. Most of the silver is removed by the first bath. Therefore the second bath always contains significantly less silver than the first one. When the fixing rate of the first bath slows the second bath replaces the first one. A new fresh bath then replaces the older second one. The two bath method works for either sodium thiosulfate or for ammonium thiosulfate.
The actual rate of removal of thiosulfate during washing is constant whether a single bath or two bath system is used. This is based on the same composition of the fixer in all the baths. What is different is that the two bath system insures that the concentration of silver-thiosulfate complex is lower in the second bath. Therefore the reduction of the complex in the paper is easier to attain. Once again it is important to understand that there are two processes going on during the wash. The removal of thiosulfate and the removal of the silver-thiosulfate complex.
There are better compounds to use as fixing agents than thiosulfate. Haist mentions several in his book "The Monobath Manual." However they are all more expensive and/or harder to obtain than thiosulfate. Which explains their use being limited to monobaths.
Thank YOu for asking some very well thought out questions. Once again the fixing process is all about kinetics. The use of a washing aid only effects the speed of removal but not the final result.