Network monitoring eats up a ton of memory (holding onto the header data for each request). It should be opt-in per-site.
In addition to opt-in per-site, I think an option to enable for all sites would be good.
Closing as invalid, as this bug pertains to the old inspector UI and/or its tests.
Please file a new bug (https://www.webkit.org/new-inspector-bug) if the bug/feature/issue is still relevant to WebKit trunk.