Dmitry Kulikov has developed a plugin for Firefox that does exactly this. You can find it here: https://github.com/dmitry-kulikov/firefox-devtools-auto-open
and can be installed from the firefox add-on store here: https://addons.mozilla.org/en-us/firefox/addon/devtools-auto-open
From my experience it does not bind to the new tab from the start. As such you might miss some network entries. I expect this is due to this being an add-on and not natively implemented.