Steamed/baked wheat buns of various kinds have been eaten in China for a long time and are known in Japan as well under the name manju (饅頭). Wiki says they were first introduced in 1341, but they're still associated with Chinese food and typically served as sweet snacks with tea, not as a meal. Savoury meat buns (肉まん nikuman) do exist, but they're nowhere near as common as bao in China.

There are also many cracker-like hard "breads" made from rice flour, notably sensei, but these too are snacks, not mains.