Here is another picture; you can see the space at the back. water pours out of it when the rainfall is decent. Probably 75% of the water gets carried away appropriately, 25% spills through that crack onto the cement and against the wall.

" 25% spills through that crack onto the cement and against the wall. "

Very well may end up in your basement or crawlspace. Will not harm the sidewalk substantially.

I wouldnt want that. I would want the water going away from the house 100 percent, not 75 percent.

I would start by raising the downspout to see if that improves the flow into the "eavestrough".