Fix Network tab not capturing requests after hot restart and clear (#9856)

* Fix Network tab not capturing requests after hot restart and clear.

Re-enable HTTP logging and socket profiling when new isolates are spawned,
reset clear timestamps using the VM timeline, and keep the search field
enabled while recording.

* Add PR links to Network profiler release notes.

* Address review feedback for Network tab lifecycle fixes.

Keep search enabled when paused with visible requests, handle RPC errors
during clear, and wrap isolate profiling re-enable in allowedError.

* Update copyright year to 2026 and refactor timestamp values in network tests for clarity. Adjusted start times in various test cases and improved code readability in the HotRestartNetworkVmService class.

* docs: remove redundant release note for Network tab

* Address review feedback for Network tab lifecycle fixes.

Use VM timeline timestamps in updateLastHttpDataRefreshTime, restore
clearData helpers and the profiling TODO, and keep the Network search
field always enabled.