Well, d’oh! 
But on the other hand the 1N4001 has a higher forward voltage so with 9×1.41-1.0=11.69 we’re back where we started 
But would they be crazy enough to design with a 20-25% no-load factor? And then tell people that they can use any wall wart that produces “about 9 V AC”. And not just “about 9 V” that I got from the PDF manual linked from that page; it turns out the HTML manual says “If you want to use another power supply instead of the one supplied, it must have a voltage output of about 7 to 9 V AC and a capacity of at least 300 mA.”
After a bit more googling for actual schematics, I found a MuffWiggler thread from 2013, where someone suspected a half-wave doubler:
https://www.muffwiggler.com/forum/viewtopic.php?t=89440
which indeed matches all the published data.