I love Rust, probably one of my favorite games ever. However, playing on a normal server is pretty much a full time job. It seems that people either move onto something else or become entirely consumed by the game. Any games that are like Rust but aren't like that?
If you want the base building and survival aspect but less hardcore you could do either Valheim or Grounded. You can limit both to servers where you just play with your friends. Vastly different games astheticaly from Rust though if that's a deal breaker.
Valheim is a Viking game where you're a dead Viking warrior cast into a purgatory of sorts and have to prove you're worthy of Valhalla.
In Grounded, you're a kid that's shrunk down to the size of an insect and need to find a way to get big again. Not recommended if you're afraid of spiders.
Valheim's boss fights veer into Souls-like territory after the first couple. Hope you like dodge rolling.
Valheim and Grounded are both great. Valheim creates procedurally generated worlds, while Grounded has a static world. So Valheim has a more natural feel which encourages you to chart coasts and biomes without exploring fully, while Grounded is highly detailed and encourages you to go over everything you find with a fine-toothed comb because there are lots of hidden things.