Personally, I would rather upgrade my headphones than upgrade my source (with the caveat that my source is "good enough"). You will see the most improvement by upgrading your headphones. I would rather spend the $1300 for a JH Roxanne than for a AK120mkii, as I feel like I would get much more improvement for my money.
I don't quite agree with your analogy. In my opinion, the differences don't matter when you are 3 inches from the screen.
I also do not agree that there is that much difference, or that the 846 is unusable on anything but a high end DAC or DAP. Shure designed these headphones to scale well, and they are designed to be driven out of smart phones (because Shure does market research and knows who buys their products).
As long as you enjoy something, you are not doing something wrong. Please do not tell other people that they are wrong because they don't share your opinions. That is why they are called opinions. Your way is good for you, but my way is good for me.
And just so we are clear, the 846 is my backup, gym headphone, so it is not like I couldn't afford a high end DAC or DAP if I felt it was justified. I just feel like, for me, it would be a waste of money when I am happy with what I currently have.