I add some speculation re. the "So what made people switch over to this particular shape?" and "Why this specific shape" parts of the question here. The answer is based on stoves and pans as used in Mongolia. Of course Mongolia is not China, but they have a long history of cultural interaction and also use round-bottomed cooking vessels a lot.
Flat-bottomed cooking vessels are great if one has a flat surface to put them on. With modern stoves that is usually a given, but not necessarily if you go further back in time. Semispherical vessels on the other hand work well as long as you have three points where to base them on. E.g. three stones, different types of tripods or stoves with an opening above the fire, as long as that opening is roughly as wide as it is long.
Re. tripods, the only free image that I found is this one:
(from Pallas, Samlungen[sic] historischer Nachrichten über die mongolischen Völkerschaften, St. Petersburg 1776, vol.1)
But there are images of a different (and I think more common) kind of tripod or quatropod here and here.
Open fire inside a house or yurt has two major problems, which are smoke and the risk of burning down your home, so people will prefer closed stoves over open fireplaces. Closed stoves in Mongolia and also rural China usually come with a circular opening in the upper surface, which is a perfect fit for a round-bottomed cooking vessel. Some images from China here, but if anyone finds a free one, I would like to include it here. This is a rather crude version from Indonesia to give an idea (source)*:
As indicated in a comment below the other answer, a round hole is obviously a good fit for a round-bottomed cooking vessel. But it is equally obvious that a round hole can be covered by a flat metal sheet, so the round-bottomed shape is not strictly necessary for being able to use a Chinese-style cooking stove. But it does not hurt either.
As an aside, stoves with circular openings in the upper surface were once common in Europe too (image source):
My guess is that a larger circular opening means that the heating power of the stove is larger (the food gets hot quicker). If that is true, putting a flat-bottomed cooking vessel on a metal sheet would be less effective than putting a round-bottomed cooking vessel onto a circular opening on a cooking stove.
*Stoves in rural China seem to be mostly made from brickwork, while stoves in rural Mongolia are overwhelmingly made from iron.