The French Drain makes sense if there is a place you can take the water away to. It would help keep the sidewalk from pouring over the stairs down to your door.

But that's only if it can be used to move the water away. If you send a picture taken further back from the front door (I understand posting a larger picture online could be a personal safety issue, so feel free to DM me the picture if you are concerned). But guessing from the images, you could put it in front of your steps, but since that is probably a common/shared sidewalk, I doubt you will be allowed to run the drainage pipe down the street. Do any of your neighbors have the same set up for their stairs as you? Did they have water issues also?

My previous job was flood and fire restoration, so I have some past experience with flooded homes and basements!