SOLUTION
I positioned the open window button in the header so that the site selection would be available in any section of the dashboard.
The selection should clearly show the cards of sites. To make it easy for the user to navigate, in addition to the name, the card should contain a screenshot of the home page and the URL of the site.
Additionally, on the card, I arranged frequently used actions.
I've formalized the various states of the site card:
selected,
paid,
trial expired,
unpublished,
blocked,
and other rare cases.
Based on user expectations, I added filtering and sorting.
For the situation when a user has a lot of sites, I added display in the form of a list.