When I go to the Site Kit Dashboard I see the same errors, “Insufficient permissions in Search Console” displayed under “Search traffic over the last 28 days” and See how your content is doing” sections. The errors are also present in troubleshoot mode. I will upload the screenshot.
As I was thinking about this problem showing up, it only happened after I moved my domains from Google to WordPress. Not sure if that is relevant, but I never experienced this issue when my domain was at Google.