I don’t think it’s a myth at all. Arabic was written with a reed pen or ‘qalam’ [0], and that can certainly be seen in the modern script (e.g. [1], to take the first random example I found). But its ‘joined-up’ nature seems to have been a regional style — it’s also found in scripts like Syriac, Avestan, Mandaic, Manichaean, and even the delightfully bizarre Bactrian Greek (there’s some nice samples in [2]).