Very nice! I really like the minimal aesthetic, and the audio was nice and crisp. Aside from the things you mentioned, I was only able to get a fuller story in exploration mode -- was that intentional?
I think a stronger indication that your time is drawing to a close would be beneficial; I felt like both the lose and win screens came up rather abruptly. I heard a growl at one point in the beginning -- the sand beast? Perhaps having more of those with greater frequency as time counts down would help.
Additionally, perhaps a slightly more forgiving hit box/collider/what have you for larger stones that only just poke out of the ground. My progress was impeded by what seemed to be inconsequential changes in height (for example, the stones that lead the path out to victory -- if you could walk over them rather than having to walk around them, I feel that would improve the experience).
Overall, though, great job!