I don't have Virgin Media Hub, but my pihole has been configured as the DHCP server for years.
The PI's Lan cable is connected directly to the router, so connectivity isn't an issue. I rarely restart it except for routine maintenance, but yes if the PiHole goes down, then everything starts to throw connection errors. This is expected. If the router was the DHCP server and the router goes down, then you would also lose internet.
However, since it's separate from the router, if you ever do need to take it offline, you can always explicitly set the IP configuration on a computer to bypass the pihole temporarily.