How to restart the wifi part alone instead of restarting the whole system?
From the Wi-Fi icon the second option is Turn Wi-fi Off. Select this, wait a few seconds, then open the Wi-Fi icon again and Turn Wi-Fi On. This is how you would restart your Wi-Fi on macOS without restarting your whole computer.

Also to elaborate on your other question(s) - there are a multitude of networking policies and configurations that could be complicating this connection. The hardware itself could be out of date or possibly just full on users and you're connecting when someone drops off - there could be network policies in place that take extra time to validate your machine. The list is quite endless. If you are eventually able to connect it's likely that you'll need to work with your IT folks to figure out the specific issues - or just deal with this buggy connection - but you shouldn't need to reboot your entire machine to make this connection work. There's really not much difference in doing that vs. turning the Wi-Fi off and on again from the menu.

Good luck!