Very valuable points, Apostolos!
What I would like to point out however is that the case of China can hardly be generalized; Didi has been an extremely well-funded company and the Chinese market is one of the hardest to entry (language barrier, cultural barrier, legislation, regulation, technology constraints and more).
If we look for instance at the taxi / personal transportation market in France or Germany, we shall see that local players like leCab or mytaxi have managed to co-exist with mega-companies like Uber and are likely to preserve, or even increase, their market share. This however did anything but stop Uber from being successful in their home markets.
My point is that this strategy works better in markets where the degree of openness is still (very) low and high barriers to entry are likely to be present for long time more (like China, Japan, Turkey, Korea, or — in some cases — Greece and the other South European markets).
Local Marketplaces have local winners, only if the respective markets are very local (barriers to entry are very high).