No. I used to own the bigger brother (DSO7104A) which is built on the same platform. The fat traces where so annoying that I used it in high resolution mode (which comes with it's downsides) most of the time. On a smaller screen it is probably less noticable but there are way better choices nowadays to get a crisp trace.
768 vertical pixels on a 12" screen = yes, you are going to see a lot of noise. You're penalizing them for giving you more Y resolution than other scopes in their class.
Again no. Even on a small screen the traces are very noisy. Put a modern, lower noise DSO next to it and you'll see the difference.
The problem you perceive is a combination of the added available resolution and the rendering algorithm they use, which takes advantage of that resolution by showing additional intensity-graded pixels that you'd normally never see at all. It does
look noisy, but the correct remedy is just turning down the intensity, rather than switching to a worse scope (i.e., everything else in the same market sector) or the high-resolution display mode (which sucks.)
The only real problem is that their default intensity setting is too high. The DSO6054A is probably the best possible fit for the OP's original request, if they are not interested in the older HPAK scopes or the TDS700D generation from Tek.
(Actually the TDS3054A would be a good fit in the responsiveness area, but they have an absurdly-small acquisition memory. And they aren't much newer than the TDS784D that can probably be purchased for about the same money.)