Step 2 of migrating away from singletons for NetworkPersistenceManager.
This stashes the NetworkPersistenceManager on global DevToolsContext,
and cleans up its instance in testing deinitialization.

Bug: 493763857
Change-Id: If33e598337d6235bb3e7fe08986b3318643d2fe4
TAG=agy
CONV=75244b23-a97f-4b3d-a0a8-4c867ebe2af7
Reviewed-on: https://chromium-review.googlesource.com/c/devtools/devtools-frontend/+/8034240
Reviewed-by: Mathias Bynens <mathias@chromium.org>
Commit-Queue: Simon Zünd <szuend@chromium.org>