Would just like to echo my views regarding Pilgrim Street.
I reckon Pilgrim St would be good for the Chesters, but I would assume most 'casual' passengers/tourists (with the Angel branding, one would assume these give the highest usage) would want to go to Durham on a whim etc. With Eldon Square being right in the heart of Newcastle (you've got Norhumberland Street and lots of other connections at Haymarket and the Metro etc) this would be where most people would want to catch it from, including me! If I had to walk to Pilgrim St (tourists may not know the way and might not be really impressed by how grim it is(!)) to go to Durham I'd simply stick to the Arriva X2 or the GNE X21 (drawbacks to this however are the relatively low frequency and reliability [using knackered presidents] as well as the average Mrs Jones not knowing if the bus serves Durham with having Bishop on the blinds).
Chester I'd assume would have most 'regular' passenger use so this, to keep delays at bay, could use Pilgrim Street as one would think regular commuters would know the way to Newcastle. This could also be used to connect with various other services, ie 56 and 27 as well as Stagecoach city services, X47 etc).
If there are any south of Gateshead, I would say running the Durham services would be priority - unless it's something really bad in Newcastle - run them straight to Eldon Square. Chesters can be allowed to throw people off at Gateshead and then run back to Chester as required however Durhams must keep going.
RE: Go North East: Service Suggestions