Early digital lines were purely circuit switched and didn’t add any jitter (unlike modern VoIP based “lines”), so there shouldn’t have been any problems running analog or digital fax over these.
V.34 was mostly a speed improvement, as far as I remember, no?
I think WarOnPrivacy was thinking of the T.38 standard. Once T.38 gained enough adoption in the various providers fax over IP became far more reliable.
edit: That is I assume that they meant VoIP when they say digital phone lines. Fax over digital ISDN lines has been reliable since its inception as far as I am aware.
V.34 was mostly a speed improvement, as far as I remember, no?