Yep, turmeric and red cabbage will also dye. They are both notorious for poor colour fastness, but that may not matter here. Other food things you can use are the water strained off black beans (mauvey blue), and safflower stamens (yellow, from Asian groceries), oh and I think sumac (but not the pickled kind - brown colour??) Avocado skins and pits, and pomegranate skins (I think), and fresh green walnut shells, and um... You can also work with rusty iron and copper to make colours, and there's heaps of European plants around in temperate English-speaking countries whose dyes properties are well known.
Kool aid is another synthetic petrochemical dye as I understand, and although I'm not sure about the specifics of this brand, many food colourings are actually chemically in the same class as, or identical to, industrial fibre dyes, with excellent dyeing properties and any attendant carcinogens, toxicity. At least in the US, anyway, which has relatively lax laws about this stuff.