The first and the last game servers are based on Unreal Engine 5 and I'm actually not 100% sure if they expose themselves properly in the SteamServer Browser, you have to understand that only most Unreal Engine 4 games and surely all Source and Gold Source are usually exposed properly.
For RUST I'm not 100% sure if it exposes itself to Steam properly.
So in conclusion not all games will show up in the SteamServer Browser and especially not in the LAN tab and it is not even a requirement from Steam that the dedicated servers must show up in the SteamServer Browser.
Sure for each game it is different but usually you will be able to join to the in game server browser and join like that, if your NAT Reflection is working properly you should be also be able to see and also join from within your LAN <- if NAT reflection is not working properly you will be only able to join from outside your LAN with your public IP or the entry in the in game server list or from within your LAN with your internal IP.
Hope that makes sense.