The Phoenix chipset seems to encompass a lot of other functions, including processing.
That is correct. But problem is that many here are discussing "scope responsiveness" without explaining what it means.
Keysight 3000T has many things in hardware, but they "windows" part, the UI, is the one that is so fast.
Rigol MSO5000 has very fast acquisition engine (not as fast as Keysight but very fast), and even FFT and other things are not slow. What is slow is windowing interface, the UI. You move something on the screen, it is slow. Same thing with R&S 3000, try scrolling horizontally with a finger...
Where all these manufacturers are not doing very good job is in the UI. Most of them seem to use some ready made windowing library, that seems to be not very good choice for real time graphical instrument..