In textbooks it is usually taught that once actual price level is higher than expected, short-run aggregate supply will eventually shift up so that new expected price level is closer to the actual price level, until AD and AS intersect at the long-run AS curve. But why doesn't the AD do the same thing? When people start to expect a high price level, they will bargain for higher income and interest rates, so they can now buy more things at any price level, that means AD will shift up also. Why such an asymmetry exists between AD and AS?
It does, in the long-run. It's partly a question of response time, and as you outline, AD has intervening steps that AS does not (wages in most markets are sticky and take time to adjust, while sticker prices in many markets can change in a day or less). In general, thinking carefully about response times will help alleviate some of the ambiguity in the models you're looking at.