On Apple’s Safari web browser on mobile and desktop the Kagi extension requires read / write access to all web pages. It seems like the intention is simply to redirect search from the limited options available in safari to Kagi. However, there are other extensions that do something similar but do not require such extensive permissions. For example, I found this extension which only needs permission for the search engine you actually have configured in iOS, then it redirects based on that. It’s asking for a lot of trust for a user to give an extension full access to all browsing activity, especially if it’s not necessary.