I believe OP is stating what we all experience to be true: powering off your phone closes the apps, including the browser, and archives your browsing history.
IMO, you can't claim you turned your phone off (i.e., powered down) but then claim the recorded 2:27 AM browser search (deleted by user per Celebrite report), actually occurred hours later or around 6:30 AM.
I'll attempt to clarify: based on my limited experience, I think if you abandon Safari to use another application (but leave Safari open in the background), and later return to the abandoned Safari search, I think it might be possible that the timestamp in Safari may not be in real time--
if the function "Background App Refresh" in the user's phone settings are NOT ENABLED.
In other words, I think if phone remained powered, and the user returned to the abandoned app (such as Safari) and inputs new search info and manually refreshes the search app, Safari would once again begin reporting in real time. I hope this post makes sense.
discussions.apple.com
Background App Refresh must be enabled in your iOS device for anything to occur in the background (while the app is not on the foreground / or if the device is locked). To enable this, just go to Settings -> General. Next, click on Background App Refresh.