Even sorcerers have to spend 15 minutes (after 8 hours of rest) to regain their spells. That's a form of 'preparation'... but obviously not the kind the DD requirements are referring to.
That's not even close to the same thing.
Sorcerers and bards cast arcane spells, but they do not use spellbooks or prepare spells. Their class level limits the number of spells she can cast (see these class descriptions). Her high Charisma score might allow her to cast a few extra spells. A member of either class must have a Charisma score of at least 10 + the spell's level to cast the spell.
Daily Readying of Spells
Each day, sorcerers and bards must focus their minds on the task of casting their spells. A sorcerer or bard needs 8 hours of rest (just like a wizard), after which she spends 15 minutes concentrating. (A bard must sing, recite, or play an instrument of some kind while concentrating.) During this period, the sorcerer or bard readies her mind to cast her daily allotment of spells. Without such a period to refresh herself, the character does not regain the spell slots she used up the day before.
Regaining spell slots is not the same thing as preparing spells.
The question then becomes whether an Arcanist's ability to choose a set of spells which they can spontaneously cast is closer to the Sorcerer model or the Wizard model. As they need to select the spells and use a spellbook I'd put it closer to Wizards and thus say they don't qualify for DD.
No the question is do they prepare spells. How they cast spells has no bearing on the prerequisites for the class. If the wizard could cast spells without preparing them then the wizard would qualify even if he still had "locked" spell slots.
However at the end of the day we both agree they do not qualify.
That said, tt might be reasonable to allow an Arcanist with the Blood Arcanist archetype and the Draconic bloodline to qualify for Dragon Disciple. Their spellcasting occupies a unique 'semi spontaneous' category which blends aspects of the two more common types together. So adding in the draconic bloodline might be a reasonable excuse to 'bridge the gap'.
Again spontaneousness has nothing to do with it, only the lack of need to prepare spells, but for homebrew rules I fully agree that could be a fine compromise.