Only superuser can edit page Template output

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Our admins can create new page page types and new page templates but for some reason they can't edit the page output. There is no way to add blocks, nor can they get the page into edit mode. Is there any way to give them "superuser" permissions or is there any way to give them permissions to edit the output?
I've checked all permissions and admins should be allowed (according to the permission settings). I also enabled the advanced permissions and rechecked but still no go.

Any hints anyone?