Team member with My Tasks set by Team Members template and the same for Global Permissions.
Team member user goes to PWA > Project Center and clicks on a project name to open the Project Detail Pages.
To work around:
Go to PWA > Server Settings > Manage Groups.
Click Team Members group.
Click My Tasks in Selected Categories.
Add "Open Project" to My Tasks for Team Members as in the screenshot below.
(If you save now and attempt to reproduce the Access Denied message as a team member, you will see this message: This Web Part was unable to load, as below.)
Back in Team Member permissions, scroll down to Global Permissions.
Add New Project under Project.
Save and close Team Member permissions.
Have the team member user access the project in Project Center by clicking the project name. The Project Details will now appear as below