These all sound like tasks a real sysadmin wouldn't attempt. File browser on a server? Use mv and you can move as many files as you want, sure you'd need a keyboard but I don't see how I would do it with mouse only. Right-click and select multiple? I don't know but it's not a feature I'd even thought of.

Cockpit is great to get a quick glance at the state of servers, but for actual work the terminal is more convenient. Appreciate it for what it is and don't expect a full desktop environment to be included.