On the Mac mini M4, the USB-C port choice usually won’t matter unless you’re using a low-quality adapter that doesn’t fully support USB 3.0 speeds. Some cheaper USB-C to USB-A dongles only do USB 2.0 power and low-bandwidth data, which can handle audio but choke on video streams. If possible, try a known good USB-C hub or adapter that supports full USB 3.1/3.2 data speeds.
The other big suspect is macOS privacy settings. On your mini, go to System Settings → Privacy & Security → Camera and make sure Zoom is allowed to use your webcam. Also, check in Zoom’s own settings (Preferences → Video) to see if your Logitech camera is selected — sometimes Zoom defaults to “FaceTime HD Camera” or “None” if it doesn’t see the device right away. Finally, try plugging the webcam directly into the Mac (via a good adapter) before launching Zoom, as macOS can be finicky about hot-plugging certain peripherals into adapters after apps are already running.