I believe the Browser stores the icon’s Favicon (if it has one) as an URL to the .desktop file. That’s why they only appear when Wifi (or generally internet connetivity) is on.
The page snapshots are usually saved as data: urls directly in the .desktop file, so they persist.
EDIT: Seems I’m partially wrong, there’s sailfish-browser/apps/browser/qml/pages/components/PopUpMenuItem.qml at b60a303790fd2c0e1fa06499a87b0e878c66fece · sailfishos/sailfish-browser · GitHub which should NOT use an URL to save the icon.
EDIT2: NO actually it’s exactly what happens here.
Also