None of the documents you have found are relevant, but well done for trying.
Firstly ownership of the land is irrelevant
The Side Roads Order reals with, well, side roads. We're looking for the principal order, which originally was
The GLA Roads Designation Order 2000as amended by
The GLA Roads Designation (Amendment) Order 2000 and
The Greater London Authority (Miscellaneous Amendments) Order 2001.
However these have almost certainly all been replaced many times over, the principal order for most boroughs is
The GLA Roads (No.3) Designation Order 2008 which abolishes The GLA Roads (No.2) Designation Order 2007, but does not designate any roads within Barnet as far as I can see.
It follows that there must be a separate order covering Barnet, which is what we're now waiting for.
The fact that appeals have been lost at London Tribunals is irrelevant, I cannot imagine anyone has previously take this point, and the adjudicator can only allow an appeal based on the issues raised by the appellant. An adjudicator cannot go digging for issues to help one party or the other, or else they'd hardly be independent.