Is there a browser extension that will show a visual indication of network activity so I know to take a closer look via the network monitor. I know - it might flash all the time, but at least I can take a closer look to assess the sites that I visit regularly.

Not browser extensions, but firewalls where you can monitor your traffic on the OS:

(TCP View is not really a firewall however, it just lets you inspect traffic, but not block it or filter it)

https://github.com/evilsocket/opensnitch (Linux)

https://www.obdev.at/products/littlesnitch/index.html (Mac)

https://www.glasswire.com/ (Windows)

https://docs.microsoft.com/en-us/sysinternals/downloads/tcpv... (Windows)