Hi Axel,
I apologize, I didn't completely finish the test mentioned in my last reply. It appears in this newly designed site collection, even the global admin has the full control permission, he still has the Deny permission level:
(Deny - Add and Customize Pages
Add, change, or delete HTML pages or Web Part Pages, and edit the Web site using a Microsoft SharePoint Foundation-compatible editor.)
Thus the customization for the site will be disabled, enabling the custom script won't help either.
Given this situation, you can either customize other site templates, like a team site or community site, or submit this feedback by clicking the button in the bottom right corner of your Communication site home page. We'd appreciate your understanding.
Thanks,
Jim