Hi, first you should distinguish dedicated server and LAN server. Dedicated server is running through hlds.exe in game folder, bacically this is how public servers work. LAN server is running through "New Game" in CS menu, this is where you supposed to record demos.
KZ-Rush LAN server can be used only on LAN server. At that it configured for recording demos, that's why by default all players who join our server are automatically forced to be spectators without persmission to join CT (recording demo rules forbid any other alive players during the demo). If you want to allow your friends to play with you, open settings menu and change "Block CT" to "Off". If you want to save settings, you can do it in the same menu. More info about this here:
https://kz-rush.ru/en/article/lan-settings.
Also in case you play with friends without recording demo I suggest you to switch Mode in main menu to "No Demo". Thus /start command won't force round restart and demo recording will not be initialized.