Please give this a read:
https://www.whnsolicitors.co.uk/newsroom/individuals/private-road-laws-uk/

It seems Southampton council don’t have a good understanding of common law, or they have curious by-laws for their particular jurisdiction. It has affected your judgement. Never assume councils know what they are talking about, or that their data is up-to-date or complete for that matter.

Roads can be adopted, or unadopted, and at the same time either publicly or privately-owned.

In this case from NSG:
USRN: 4600057
ABELS CLOSE, FELTHORPE
Classification: Neither M, A, B or C
Responsibility: Not maintainable at public expense, Private Street Manager.