Renewal not recording activity
I believe the issue to be as a result of CiviCRM seeing the payment as being a test.

Essentially, I used the PayPal sandbox configuration for the "production" processor. Looking back at the contributions for the membership, it stated the type was 'Member Dues (test)'.

After setting up and using the dummy processor, the contribution stated was just 'Member Dues', and it then recorded the contribution and membership renewal activities.