Cannot edit any FSE theme through site editor

Hello all,

Our website exists since a long time. It has automatic update and is therefore in 5.9.1.

In order to modernize it, I would like to try a block theme with FSE.

I have tried to activate either Twenty Twenty-Two or a less classic one Framboise.
The display is successful but whenever trying to edit a block-based theme through the Full Site Editor (/wp1/wp-admin/site-editor.php), an error is raised:

“The editor is unable to find a block template for the homepage.”

The full error is:

Error:getHomepageParams`: HTTP status error, 404 Not Found
at https://www.couleurs-cactus.fr/wp1/wp-includes/js/dist/edit-site.min.js?ver=403e01f2b098b6a656118a51787581cb:2:121078
at async https://www.couleurs-cactus.fr/wp1/wp-includes/js/dist/edit-site.min.js?ver=403e01f2b098b6a656118a51787581cb:2:120983
at async zr (https://www.couleurs-cactus.fr/wp1/wp-includes/js/dist/edit-site.min.js?ver=403e01f2b098b6a656118a51787581cb:2:120742)
at async qr (https://www.couleurs-cactus.fr/wp1/wp-includes/js/dist/edit-site.min.js?ver=403e01f2b098b6a656118a51787581cb:2:121890)`

When openging the Developer tools tab I have a 404 error on the following URL:
/wp1/?_wp-find-template=true (fetch from edit-site.min.js).

I have tried to deactivate all plugins without more success.

Any help or hints on what’s happening would be welcome:)

The page I need help with: [log in to see the link]