At the moment you can click the "Make this my start page" link in Tableau server.
Unfortunately this link is not site specific and can be very content specific.
In other words if you set your start page to be simply workbooks or views - no problem. On another site clicking the start page will also default to workbooks or views.
However if you set your start page to views in project X and on another site project X does not exist, you will be directed to an empty page. If you set your start page to a specific workbook and that workbook does not exist on the other site, you even get an error message to say the workbook does not exist or you do not have access to it.
This is very confusing. I like that you can set a specific project as your start page.
I would like it if I could set a start page per site.
----> Community Manager Update: This idea was 'Released' with 2019.2 release of Tableau
Special thanks to Sean Li and team for their hard work on this feature - and thank you to everyone who commented and voted!
Download the latest version of Tableau here: Tableau Desktop | Tableau Software
-- Amanda Boyle, Tableau Community Manager