It's getting down to semantics, but I read it as the Factotum counts as a level 19 Wizard only for the purposes of said ability, and spellcasting is a different ability than spellbook acquisition.
Correct, but you still receive the spellbook of a 19+ level wizard, which was the point.
The wizard does not have the ability to obtain a free spellbook during play. Rather, the wizard begins play
with such a spellbook, so you're only getting the free spellbook if you activate the ability before starting play. Similarly she only gains the new spells when gaining a wizard level, so you'd have to have a spellbook, activate this ability, and gain a level while this ability is active to benefit from that access, which may or may not be practical depending on how your DM treats leveling up.