The only
background01.vtf file in the new as well as the old Steam installation is this:

... it has nothing to do with the game. But everything with the various Steam backgrounds (i believe it is used by SDK mods -the most recent i played was Black Mesa).
By mistake, the offline mode thinks it needs that particular picture at the start of our game.

In the new installation (latest Steam and after the game's conversion), it is located at steamapps\sourcemods\BMS\materials\console
In the old one it was in steamapps\sourcemods\half-life 2 update\materials\Console
But it doesn't matter where it is located; it matters why
only in offline mode it is being called, and how to make Steam stop doing that.
And even before that: why is it that
-only in offline mode- at the start, the intro videos don't show up (white screen with black bars -so at least the empty spaces have the correct aspect ratio), and it is after the blanked out intro's that the game crashes completely with the [Can't find background image 'materials/console/background01.vtf'] Error! message.