BA is incorrect.
The only restriction on surface sectors is on intercontinental surface sectors:
(4g) Intermediate surface sectors are permitted at the passenger’s expense. Transoceanic surface sectors between TC1-TC2 and TC1-TC3 are not permitted.
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(TC1 = Nth & Sth America; TC2 = Europe/Middle East & Africa; TC3 = Asia & Oceania)
As you said, surface sectors do count toward the 16 maximum allowed sectors in total but not towards the maximum number of sectors in a continent - that rule explicitly refers to flight sectors:
4(h) A minimum of 3 and maximum of 16 segments, including surface segments between any 2 airports, are permitted for the entire journey. Free flight segments within each continent are limited as follows:
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