Because of the typical required slope away from the house drainage, the walkway will always wash due to 1) rain over the path 2) roof rain if the gutters don’t fully run along that side and 3) because you’re drains-pouts are NOT connected underneath and are NOT shooting/exit/daylighting out towards the street/storm drain (if that’s what your jurisdiction does).

Note sure what is AFFORDABLE to you, but these are my recommendations from $ to $$$ which all encourage minimizing the square foot of the area of the aggregates: 1) Add more per-cast concrete steppers, 2) Add more pre-cast concrete steppers and make adjustments on the gutters and drain spouts so it’s connected and exits/daylights to the street and not the side, 3) pour concrete with the basic broom finish on the entire side just like your neighbor.

Good luck! If you choose any of the recommendations, I’d be interested in seeing after photos! Be sure to tag me!

Your downspouts may also be adding to the flow on the walkway when there is a strong flow that won't all drain into the ground there. I would reposition those. Also, we used to have a problem with our driveway flooding. We removed the hard surface and made it a gravel driveway instead which allows the rain to drain into the ground below. No lakes anymore! You could easily do that and put the stepping stones on top of the gravel.

Roof line is very broken and the down spouts are where they are because of the architectural design of the roof. Nothing can be done about the placement.