Would Sunderlands profile as a city be raised if it was the Capital of County Durham or stay as a district in Tyne and Wear.
I don't think it would make any difference in terms of how it is perceived by the rest of the country. Lancaster is the county town of Lancashire, what does that actually mean beyond its historical significance? Personally I would prefer to differentiate from Newcastle but the cat is out of the bag now, I do think there are historical and perhaps even identity differences between most of County Durham and Northumberland plus the settlements on the south bank of the Tyne.
Durham is the capital of County Durham. York is the capital of Yorkshire. Lancaster is the capital of Lancashire. Historically hugely important cities which have been outgrown in size by their neighbours. FWIW, I refuse to use Tyne and Wear in correspondence. I’m from Sunderland, County Durham!