The HD was fully formatted. It was clean Monterey 12.2 install and clean Fusion installation. No previous Fusion version on the machine.

I reviewed the install log and figured out that Install Helper was supposed to be installed to write protected directory. Disabling SIP allowed it to be installed and launched. After that I've found that I cannot share USB devices unless I disable File protection "csrutil enable --without fs". If SIP is fully enabled, then no USB devices appear in USB & Bluetooth pane. 

Unfortunately, I cannot try any other solution at the moment. I'm rebooting the machine only if it is absolutely necessary.