People have been using membership plugins for their premium content for a long time. They do have many fancy features that may not be needed if you just want to host a private family or staging site. My Private Site for WordPress can help. It is a handy plugin that you can use to take your site private to share private content. It does give you the option to keep your homepage public and hide everything else.
This plugin works with multisite networks and custom login pages. Worried about AI crawlers grabbing your content? This plugin automatically adds meta tags and headers to your website. You can also block GPTBot. The neat part is this plugin does not hide non WordPress pages, so you can build a public presence around it. With Block IP, Advanced AI Defense, and Selective Content, which are premium add-ons, you can further customize your website.