"Drupal already installed" Message on Restored site
Seeing the install page on an already installed Drupal site usually is a sign of problems with your database connection before the request has been processed by PHP.

This may have a wide range of causes, most likely related to your database server configuration.

The first thing to check is your database logs, which may give a hint on how to solve your issue.

We often experienced a similar behavior on sites that heavily use the database. E.g. sending big queries when rebuilding page caches on complex pages. The solution there was to ensure a high enough max_allowed_packet setting. See here for more details on how to do that.