Redmine

if issue goes to person who let the issue on him and do nothing, nobody is responsible for that issue to push the issue forward.

You can't solve non-technical problems by technical means...

I think it is issue management flow problem, and main redmine job is help solve preject/task management. This is issue/task management, and in business there is many important jobs what have to be solved more intensively. In that case the person which is responsible for issue have to create spreadsheet table where he wrote issue numbers which he have to periodically check. I think this should be solved directly by redmine not by another tools.

If we do custom field type user, it is not possible to do query on all projects to filter issues which is SUPERVISOR set "me"

Do you just need a "supervisor = me" filter on the global issue list? I think it's a feature that can be added.

Yes it should be enought. But I am not sure how you want to connect custom field type list, which is list of strings with "me" what is user id.

I can add this feature if supervisor is a user custom field, not a list field.

But user custom field is not possible to add as filter for global query. Maybe becaouse there is different users in different projects.

Yes, that's why I propose to change that and allow to filter with "user custom field = me" on the global query.

Yes, it will be enough.
Now we extend supervisor to possibility select multiple supervisors, in some tasks there is necessary to have more supervisors in state of projects where is more levels of responsibility. As I see in user custom field there is possible set multiple checkbox config.