Leamside falls into one of those categories of being so useful, it's almost too useful, it would create too many options. Each option has different competing groups putting forward ideas as to what it should do, as exhibited in the comments so far.
Those that live south of York see Leamside as a way of getting freight off the ECML so they can get from York to Newcastle as quickly as possible, without the need to spend billions extending HS2/NPR north of York.
If you live in the Tees-valley area, Leamside (along with Stillington) offers the possibility of linking Newcastle to Middlesbrough with a regional express service (either via Leamside or Durham), and local stopping services, that avoids the longer, slower Durham Coast Line or a 25-30 minute journey west to Darlington (before you even start to head north).
People living between Ferryhill & Newcastle see Leamside as an opportunity to create much needed local services for areas that are growing commuter towns.
Upgrading Northallerton-Stockton, Stockton-Ferryhill (Stillington) & reopening Leamside creates a massive amount of extra capacity between York, Teesside & Newcastle. The key question is how best to use that extra capacity, and what takes priority, Long distance traffic, regional traffic or local traffic or some combination of all 3.
If Leamside was to be re-opened I would hope that it was rebuilt to a higher standard than it's previous incarnation (Increased speeds, electrification, a grade separated junction at Tursdale, a passing loop or two. etc). It might cost a few bob more, but we have to be realistic and accept that HS2/NPR as a new modern route on a new alignment north of York is not going to happen any time soon, so "IF" anything does happen with Leamside, it's going to have to see the North-East through the next 70-100 years.