Which Arduino microcontrollers are able to drive a parallel 8-bit ILI9341 display at 320x240@60fps with DMA or extra CPU core or both and how?
The problem is the whole concept of running an ILI9341 TFT screen at 60FPS from a microcontroller. You can't. It's just not fast enough.

To get those kind of speeds you need to directly interface the TFT panel with the microcontroller. And that takes a special microcontroller with a TFT interface built in.  As well as that you need the memory to store the image that you are displaying on the screen.

There are microcontrollers available that can do it quite happily.  But they aren't in the "maker scene". Simply because they aren't a common thing to do.

For many years now there has been the PIC24 "DA" range of microcontrollers with a built in GPU and TFT interface - but there is no free C++ compiler, so no Arduino support - so no easy route to using them for beginners.

There are now, though, the PIC32MZ "DA" rage which are more accessible - though still very hard to use.  These are much more viable for the kind of thing you want, with 32MB of built in DDR, a TFT panel interface, and available in a 144 pin TQFP chip that is possible for an experienced maker to hand solder.

But the barrier is still software.  While chipKIT has full support for the PIC32MZ "EF" range of chips the "DA" range is not yet supported.  That's not to say you can't work with them, just not with the Arduino API and IDE.

The more common way to make a games console like device is to take an ARM SoC and interface it with a TFT panel directly. The simplest way for a maker to do that is to put a TFT on a Pi and install Retropie.  

And we're not against open discussion as people. It's the site. It's not a site for open discussion. It's a question and answer site. You ask a technical question that you need assistance with, and we tell you the answer. It's not a forum for general discussion.  As such, this question will be closed as either "top broad" or "primarily opinion based". However your question does deserve an answer anyway.