Sounds like a metallic component slightly lying at another metallic part. Propably a metal-sheet.
The vids are difficult to analyse.
Try looking under the bonnet at the passanger side. The may be heat shields or similar that rattle at the carframe, up left in the corners and depths of the plastic-jungle.
Then take out the glovebox and try to take out some other parts of the dashboard.
Look for connectors or other things that might be the cause.
Perhaps its the thingies in the glovebox ? 
Next idea is the plastic in the wheel arch trim passanger side.
All screws at place, does it sit right in the carframe/wheel arch ?
The part with " it stopps when swerving" may hint at something "heavy" hanging at another part of the car and when when the car turns left/right it gets pulled off or straight on to the carframe or plasticpart of the dashboard.
Maybe a control unit somewhere in the depths of the dashboard.
Or:
Maybe the plastic-cover of the windshield/wipers rattles at the frame/glas or something else und through steering it stops because the frame winds itself.
Seems to be a good guess, the more I think about it.
I fear you wont get it, without pulling the dashboard or other parts apart.
Good for you, that Toyota has the dashboard constructed in a very mechanic-friendly way.
...I´m always amazed about my good english
Herman the German 