We have cases where "Group Site Admin" mode is used only by a group of administrators who must also have the admin mode active to perform the task. This means, they need to always ensure that the "Group Site Admin" is checked after they are logged in.
We can provide a "Quality of Life" improvement for such users if we have some sort of permissions akin to "Bypass content access control".
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Add a permission "Bypass group_sites access control" or "Always on group_sites admin mode", and use that permission to check if the admin mode is active.
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(FWIW) Possibly not dissimilar use case, but we have the users with an admin role being able to log into additional admin domain which I ended up fixing in admin mode https://github.com/localgovdrupal/localgov_microsites_group/blob/4.x/src... The whole domain is then customised to make doing admin tasks easier.