The problem using peroxide on wool or other natural fibers is how strong of a mix you use and how often do you clean the fiber.
A 3% solution of peroxide on wool that is just on the surface of the fiber that will be padded off and dried in 20 minutes is different then soaking the rug with a strong mix of peroxide.
Magic wand sells powder oxygen made just for wool rugs------you need to learn about products that have buffering agents
Powder products like sodium percarbonate or sodium perborate are different than liquid peroxide on the wool and other natural fibers.
You will have to find out what the % of peroxide is in your mix and what concentration you have when diluting product.
I have not had any problems with encap products boosted with peroxide on natural fibers when I KNOW the mix is 3% peroxide or less.
Surface cleaning is not the same as soaking fiber with peroxide but repeat cleanings will change the fiber on wool and also can damage the fringe.
Reducing agents are safer for wool and cotton when you are trying to restore a problem or stain on natural fibers
LINK to video
That stain was removed with a low ph product and an oxygen reducing agent.
Strong peroxide mix would work faster but when I have tried that on fibers it can leave the fiber gray or worse bleached out---very touchy with natural fibers.
Some wool rugs are not a problem but you need to test each one and check the dyes on the rug too to see what your product is going to do.
Less strong products and mixes------- and you will have no problem.