Short answer: apocryphal is routinely used to mean not [convincingly] true.
Although the concepts are intertwined, the definition of apocryphal denotes authenticity not authorship:
1 (Of a story or statement) of doubtful authenticity, although widely
circulated as being true:
The connotation of authorship is rooted in the ongoing theological dispute about the biblical Apocrypha:
late 14c., neuter plural of Late Latin apocryphus "secret, not
approved for public reading,"
from Greek apokryphos "hidden;
obscure," thus "(books) of unknown authorship" (especially those
included in the Septuagint and Vulgate but not originally written in
Hebrew and not counted as genuine by the Jews),
from apo- "away"
(see apo-) + kryptein "to hide" (see crypt).
Some early Jewish and Christian leaders demanded that the followers of their sects avoid the Apocrypha, because they were deemed less trustworthy than their canon of scripture. Faithfulness to the sect's unique truth claims was the primary concern, but authorship was posited as the most important measure of veracity.
In the extended use of apocryphal, authorship does not always rise to the same level for determining authenticity, but a doubtful source always casts some doubt on the reliability of any truth claim.