Membership renewal start date doesn't update?
If you check on yoursite/civicrm/admin/member/membershipStatus?reset=1 and for the status "Grace" there is a Yes in the column headed "Member" then this is expected behaviour.

CiviCRM does not have a "renewal start date".  The "start date" is the date the latest continuous period of membership started on. For any member with a status that has a "Yes" in that "Member" column a renewal will extend the end date of the current membership and not change the start date.

If you want anybody who renews after their membership end date to get a new start date then you either need to edit the Grace status and untick the box Current Membership?, or disable the Grace status entirely if it is not needed for your business rules and just have a membership status of Current followed by the membership status Ended.

You can read more about Membership status rules  in the Defining memberships chapter of our User and Admin Guide