Right now the host of the game is effectively using their PC as a server for that network game. In theory you could already have a machine set-up which is available 24/7 and has specific rules in place that mean progression only occurs once a certain number of players have continued etc. Switching to a private server to host the game would incur a lot of developmental changes. The best comparison I can think of right now in gaming terms would be set-ups like Rust, Conan Exiles, Ark and Minecraft among others. A perpetually available private server does come with costs though, a quick search online suggests that typically you're looking at anywhere from $5 - $25 with most other games that offer this as a service. It's important when logging a feature request that I do understand it correctly, so am I right in understanding that this is the kind of set-up you're advocating for @DaniFito?
If not, could you explain perhaps a comparison that is already established or provide more details on this.