I would say your phone can do that, it's just that you did not enable HDR.
The thing is, because of how our eyes and brain work (huge dynamic range plus postprocessing), the actual viewed scene was probably closer to the second photo than the first, but with a higher contrast (the camera HDR emulates the eye but you also need a huge dynamic range display to show it right, otherwise it's "compressed" to low contrast). So you can choose the contrast (first photo) or dynamic range (second photo) but not both.
My comment came out sounding somewhat mean, which wasn't my intention, but the truth is that I'm not a big fan of HDR and related artificial effects. These effects don't look anything like the imagery perceived by my eyes and brain; they're just different ways to ruin a good photo.
Once dynamic range is lost by rendering something on a phone screen or computer monitor, you can't recover it without making the image look weird and artificial. It's like mixing 24-bit audio for playback through an 8-bit DAC. You can play all sorts of games with dynamic range expansion, but at the end of the day you've done a bad thing.
Then there's the fact that a lot of problems people try to solve with HDR are actually gamma correction issues.
Of course, I may just not be taking the right drugs...
That's not very accurate because the colors in the clouds were inverted in this photo, but you're right that reality is a bit more contrasty, though not as contrasty as we think (nowhere close to the "after").
The bottom version... OK, my phone can't do that. If it did, I'd take it to the Apple store to see if they could fix it.