Currently extensions are not allowed to load on iOS when the background persistent key is set to true.
To allow a single code base between iOS and macOS, it's very welcome to allow the persistent key to be set to true and accept it in iOS. As currently Apple doesn't want to support persistent background pages on iOS, Safari in iOS can simply ignore the persistent key and run the background page as if persistent is set to false.
See also feedback assistant report: https://feedbackassistant.apple.com/feedback/9676503
