Yes, this scope has 1Mpts of memory depth, so at 1 Ghz it will be able to store 1 ms worth of data. So, this would not fill the full screen on your first picture. And processing 1 M points each time is also resource intensive, so update rates would be abysmal.
This is why deeper memory and independent setting of a sample rate is important. I think even DS1054Z can do that. I remember capturing frames worth of data of a video signal with no issues.