The phrase "overriding importance" means that something is extremely significant or crucial, to the point where it takes precedence over everything else. Full definition
«Simply restricting one food or beverage will not override the importance of a common sense approach to children's diet and overall health. (australianbeverages.org)
Personally, I see Hirst's work as a simple reflection of modern society which attaches overriding importance to «news value». (visual-arts-cork.com)
Social media outlets gave overriding importance to issues we could never have thought so important. (blog.politics.ox.ac.uk)