I've posted a route up for Pittsburgh to Grove City. I live in Beaver Falls, which is actually along Route A. If you or anyone else is riding this direction, feel free to message me. I've got a spare guest bedroom that anyone is welcome to crash at.
Here's the link for my route:
http://www.bikely.com/maps/bike-path/Pittsburgh-to-Grove-City
There are three main differences compared to route A. First, much of the ride along the Ohio River is on the east side, following side roads that run parallel to 65. This is a good route that has slow traveling cars, except for a few spots up near Conway. Unfortunately, the side roads end there and you have to ride through some parking lots and walk or ride along a sketchy sidewalk for 1.5 miles. I'll admit, it's not any fun at that part, but it sure beats riding route 51 there.
The 2nd big difference from Route A is that after Beaver Falls, my route continues north along the Beaver River. Route A will take you up through Ellwood City into Portersville on 488, which is a miserable hilly road with cars traveling way too fast and no shoulder. My route mostly goes along Old Route 18, with a few stretches on Route 18. I ride this route regularly when I go to New Castle, as it's a nice quiet rural ride with very few hills.
After getting to New Castle, you'll head east. While I haven't ridden this road yet, I imagine it's pretty quiet since it parallels 422 and all vehicles would be on 422 except for local traffic.
Finally, the route parallels Route 19 on side roads such as Studebaker St. I've ridden these roads up to Grove City and I believe a total of 3 cars passed me. I have no clue why Route A says to go on 19 when there's much better roads just a mile away, perhaps it's just easier to say 'Follow 19'?
I hope all of this helps and feel free to ask me any questions. If you're following Route A from the south, I imagine you'd pick up the Montour Trail before McDonald and ride that into Corapolis. Once you reach Corapolis, you'll be good to follow the route I posted for the next 60+ miles.