It appears that the roads that Yorkie used existed by 1850 according to historic mapping. Though I'm sure those who could not afford a horse walked on the roads, I suspect the main purpose of Denby Dale Road was for long distance horse-drawn traffic while Ings road was for such traffic and others to bypass the busy streets of Wakefield, as well as provide access to the newly constructed property on that route.
I suspect that at the time, little thought was given to pedestrian traffic because the roads had very little property at the time and horse drawn traffic did not suffer excessive problems mixing with pedestrian traffic.
I suspect pavements were subsequently added to both roads at a later date, but cannot verify when. The public park housing the Premier Inn was constructed around the 1900s, and Denby Dale Road appears to have been widened with property added, so this may be when a pavement was constructed.
Both roads appear to have been widened to dual carriageway in the 1970s. I suspect Yorkie's legitimate complaint is with the dualing and lack of pedestrian crossings. I would agree that the post-war transport policy and urban planning, much of inspire by the 1960's Buchanan Report, was misguided and urban dual carriageways inspired by this should be redeveloped to better support pedestrians, cyclists and public transport uses. Installation of pedestrians crossings, as well as replacing roundabouts and subways with at-gate traffic light junctions play a part in this, as do bus lanes.
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For what it's worth, since people have commented on Wakefield, I think I'd add that it's a fine city which suffered mistreatment from (actual) Nazis following by unsympathetic demolition and somewhat misguided redevelopment. Many problems remain, but with state and private investment in the city, careful planning and good architecture, I believe the city can grow and prosper.