Between 1838 and 1938 there were only six records of Sunderland Brocketts in St Catherine’s Index of births, marriages and deaths in England and Wales: five marriages and one death. The marriages were of James in 1852, Richard Hodgson in 1867, Frederick in 1868, Robert Edwin in 1900 and Sarah in 1910. The death was of Robert in 1906.
Apart from Sarah, all of these marriages could have been Brockett lads from elsewhere in Durham or Tyneside marrying Sunderland lasses. I don’t think there are any Brocketts in Sunderland nowadays.
My guess is that your Sunderland family was the only Brockett family there, perhaps descended from Newcastle Brocketts, or one or other of the families near Whitby or Hartlepool. Do you know of many descendants? (I guess you know that Bishopwearmouth is now a part of Sunderland.)
A generation or two earlier there is an administration dated 1742 of Jacob Brockett, son of Robert Brockett of Sunderland by the Sea, Blacksmith.
Ultimately, I suspect that they were part of the ancient Yorkshire Broket clan, some of whom moved north into and through Northumberland in the service of the Percys and on up into the Scottish lowlands, Lanarkshire and eventually Glasgow.