Carmine is better known as Red 4 these days. Doesn't have much taste. Saffron adds basically no taste in the amounts typically used for something like Saffron rice. Squid ink again, mostly for the striking color. The taste isn't particularly great.
Turmeric can go both ways, but the ground turmeric that's historically common for preservation reasons is much less flavorful than the fresh root. It's mostly a color thing.
Of course, we can also just open up a medieval cookbook to see what they say. The Forme of Cury is a nice 14th century example that's available from Gutenberg:
https://www.gutenberg.org/ebooks/8102
As to colours, which perhaps would chiefly take place in suttleties, blood boiled and fried was used for dying black. saffron for yellow, and sanders for red. Alkenet is also used for colouring, and mulberries; amydon makes white; and turnesole [for yellow]
Alkanet is commonly used today for Rogan josh, but historically would have been more known for rouge and dying wine. A Mediterranean cookbook might have instead chosen amaranth for the same purpose
You need so little tumeric or annatto to color food and they impart so little flavor in so many applications that the reason they are used is very obvious.
