Q. Would it be better to check all the staff in my organisation, just to be on the safe side?

No.  It would be inappropriate, and, probably, an infringement of someone’s civil rights, to apply for an enhanced level check when the person does not have any access to vulnerable people.  The best way to protect children and vulnerable adults is to improve management and supervision arrangements.  Ideally, there should be no secrets within the organisation about who has had a check and who has not.  Which roles require such checks should also be clearly understood by managers, etc.