As for the “why is flat better for mobile?” question, I’m far from a design expert, but I think this article puts forward a good argument for it (and explains it better than I ever could). Long story short, flat design allows for greater contrast and more flexibility. When you’re working with a smaller amount of space, gradients and gloss can be unnecessarily complicated and distracting. It’s the same reason why IOS and Windows dropped their old glossy designs they had for their operating systems several years back.
I’m not against the old peacock or anything, but on the subject of NBC News, their visual presentation is looking rather dated IMO (although not nearly as dated as ABC). I’m sure they’ll eventually adapt their graphics to suit the new peacock, but I doubt they’re in any rush to do it given the similarities between the logos. I can be nit picky about this stuff myself, but I honestly didn’t even notice the changes when the new promos first aired. That’s how subtle they are IMO. But to each their own.