Yes, lots of people live along Honor Oak Road - just as they live along London Road and the rest of the South Circular - but these are nonetheless main roads, both historically and in the way they are now designed, with traffic lights, pedestrian crossings and bus routes etc along them.
Honor Oak Road is not historically a main road. It is a narrow, twisty and hilly residential Rd. The P4 is designed for such roads.
Forcing its use as a main road is the underlying mistake that causes many other problems.
The road that historically connected Peckham to Sydenham perhaps back to the roman times is Wood Vale which is mostly wide, flat and straight.
But the residents on this road are even richer than those on Manor Mount and have long ago successfully repelled through traffic and bus routes.