Limiting permissions - global areas

Permalink
I've been wrestling with global areas and permissions. The site will have "simple-admin" user group (and users) who have access to all areas, except to global areas.
I haven't activated Advanced Permission - I would like to keep things simple, if possible.

I'm sure I managed to make this work with another site, but not now.
I have two user groups, admin and "simple-admin". Both work otherwise well. The simple one has permission only to certain parts of dashboard.

- gave Site Access (dashboard / System / Site access)
- gave File Manager permissions (dashboard / System / )
- gave permissions to add certain pages ( dashboard / Pages and Themes / Page types)
- gave several permissions to sitemap, pages etc (dashboard / Sitemap) /
AND here tried to give permissions only to some of the stacks. But it does not work. The global areas are still "open".

Tried giving all permission to stacks back, then removing again
created new user group - and gave permissions
created new users
updated to 7.5.2 (from 7.5.1, got desperate)

Forgot something? Did too much? Something wrong with database?

PaiviK