"Chlorine Bleach does not yellow clothing...."
Oh yes it can.
Sodium hypochlorite bleaches outright attack spandex, Lycra, silk, wool, causing among other things yellowing.
For cotton and linen fabrics "chlorine" bleach can cause yellowing via formation of oxycellulose. This occurs when too strong a solution, and or at too high temperatures, and or chlorine bleach remains in textiles that are subsequently subjected to high heat (clothes dryer and or ironing). The reaction is a response to a tendering of the fabric which produces the yellow shade.
Fabrics treated with sodium hypochlorite bleaches also can yellow if exposed to ultraviolet light such as rays from sun.
Chlorine bleach is difficult to near impossible to rinse out of fabrics. Cotton and perhaps linen especially have an affinity for the substance. Hence in commercial/industrial laundries several rinses follow the "bleach" bath. Then a final sour/anti-chlor rinse to ensure all traces of bleach are removed.
OTOH the standard one rinse (most American top loaders) or even several in a front loader may not cut it. If you can still smell chlorine bleach on textiles, it still remains.
There is a reason why European laundries and households long have avoided "eau de Javel", chloride of lime, chlorine bleaches for laundry use. It harms textiles in the long run. Some situations have no choice, such as hospital/healthcare laundries who aren't using thermal disinfection methods. You do find chlorine bleach in say France and other parts of Europe, but in the household cleaning supply section, not laundry.