Primary secretaries are listed as being members of Primary presidencies
I understand it's easy to get confused about Presidencies, I know in some places Secretaries are listed as members of Presidencies (or the Ward Clerk as a member of the Bishopric). However, GHB 12.3.2 and 12.3.3 list the Primary Secretary as outside the Primary Presidency, so I expect the Access Table listing of "Primary Presidency" does exclude the secretary.
The secretary being outside the presidency is also reflected in that when a Primary President is released, only the Counselors must also be released (even if they're going to be re-sustained to the exact same callings), while the secretary remains unaffected (unless her calling is also changing, of course!).
Of course, the question of whether the Primary Secretary
should have access to the class rolls, is a different issue, and I agree that the answer is definitely "yes"! The only way to push for this to change would be to submit feedback on any page in LCR.