I think that the issue was not color transfer from colored clothes, but rather that you can use bleach on the whites. You don’t want to bleach the colored clothes.

I don’t bleach whites because I don’t care about having pure white whites, but…

googles

https://www.homesandgardens.com/kitchens/how-to-use-bleach-in-laundry

How to use bleach in laundry – expert tips for the whitest whites

These expert tips on how to use bleach in laundry will get your whites gleaming