If a VPN makes the game work. Then the connection issue causing the black screen is most likely being caused by either the home router or ISP. A VPN tunnels through both of them.
The easiest way to rule out the home router being the issue, is to connect the PC directly to the internet modem. Then power cycle the internet modem so it assigns a new IP address to the PC.
If BF6 can connect now, then the home router is the issue. If BF6 still doesn't connect then the ISP is the issue.
If ISP is the issue, the next step would be running a traceroute to Google.com to see what the ISP network looks like.