When I log in as one of our users from a desktop browser and click the "View All" button on the Activities (timeline) component, I get the following error message:
You can't view this page, either because you don't have permission or because the page isn't supported on mobile devices.
The user has "Access Activities" permissions.
Best Answer
Go to the (Classic) page layout used on the Lightning page and add the "Open Activities" and "Activity History" related lists to the page.
Apparently, then--to some extent--the Activity component on the Lightning page relies on the Classic page layout to determine which activities can be exposed through it.