Amount of information in scaled-down images
Averaging-and-truncation certainly loses more information. It is not because you are not preserving any specific pixel: it is because you are taking an even probability distribution and giving it a strong bias in one direction.

If I have understood you correctly, you have taken the 16 possible 2x2 images and transformed one of them to white and 15 of them to black. This 15:1 split is about as bad as you can get! It destroys information, and it is the reason why your results are so bad compared to the "take top left corner".

The nearer you get to an "8:8" split, the less information you will lose. So try averaging and rounding: what I think you call "taking the median". This will split 11:5, which is less bad. It is still not very good, because the 6 cases of "2 black, 2 white pixels" all have to be translated either to black or to white - that is why you can only get 11:5 rather than the ideal 8:8 between black and white outcomes. [It would be interesting to try "use the majority value if there is a majority value; if not, use the top left"].

If you try $3\times 3$ blocks then either a clear majority of pixels will be white or a clear majority of pixels will be black: a straight 256:256 split. This is the best possible outcome, and if you simulate it then you should find it leads to the least information loss.