A Park and Ride at Follingsby could be a strong benefit of this route. At the moment there's no real P&R options for Newcastle south of the Tyne. I'd have thought that site - just off a quiet motorway junction, away from the traffic on the Newcastle western bypass, with few contradictory traffic movements, and access to the A19 - would be perfect. Heworth is OKish but you do get clogged up both at the A194(M)/A184 roundabout and then with traffic heading into Gateshead/Newcastle: when I've used it during commuter times it can easily take 10-15 minutes to get from the motorway to being parked up at Heworth, which significantly reduces the time savings of that route.
I'm less convinced by the South Hylton - Washington extension. The old line is a popular and useful leisure walking route, on the edge of nature reserves, which will stand against further development. Presumably, that's where a lot of the cost comes from as well - as mentioned upthread, it would bring the challenge of having the viaduct as part of the route and subsequent maintenance of it to railway standards, and of constructing the line under the A19.
A stub extension to Washington would be much cheaper, easier to do, and relatively well used I'd have thought. You'd like the route to be closer to the town centre but the railway is where it is and Washington would be no different from many other towns and cities in that regards.
