Yes! Onward!
@SushiRolag A drawing really makes you think about what you need to do and exactly how you will accomplish it. If you don’t come up with a cotton spinner here, I have a list of resources you could check out. @SpinsterJulieB is right, the existing thread around the damaged area is suspect so this would give you enough to work with. Using tea is a perfect natural dye for cotton. I did this https://www.apartmenttherapy.com/how-to-dye-textiles-with-coffee-tea-10843 with white cotton flour sack dishtowels for that vintage look and don’t think the process would be much different for yarn. Maybe just try a little bit first. Don’t forget to wash & dry your replacement pieces before inserting to account for shrinkage.
@Carlota That rag rug will have new life! It’s probably more scary than difficult.