What I really like about the way Wild Apricot does it now, is if there are any pending membership payments or applications, the user is taken to their Member Profile page where they are told about that pending item.
If you do add the ability to customize where the logged in member lands, I think you should either force members with pending items to have to go to Member Profile page or at least give us the option of having them land there when pending items exist.
I think the ideal long-term solution would be a member landing page that is a member dashboard. All all key membership details are displayed including past/future event registrations, outstanding invoices, current membership status, renew button, change membership level button, past donations, contact information (editable).
Kyle Mitchell | www.spec.bc.ca