The overall feel is nice.  However, there's not really any point to making the player hold down a button throughout playing the game. This could even make your game harder to play for some people (if they have a physical disability of the hands) without adding anything to your game that would justify it. It also has the thematic and contextual issue that this character who's about to retire doesn't know the layout of the neighborhood (implied by the player having to find out each playthrough).

I tried it out twice. On the second time, the game eventually handed me a letter for 9 despite there only being 8 houses that I could find.