If you replace fast movement with an archetype can you still use Flying Kick?

[Zautos']

So for example, if I play an unchanged Flowing Monk.
that replaces fast movement can I still use flying kick?

Also, I know that this archetype doesn't work with the unchanged version. But that's an easy fix.

[James Risner]

If you don’t have fast movement from monk levels, you don’t have a defined fast movement.
You still can’t use if you have fast movement from barbarian levels, because things are written single classed.

[avr]

unchanged = unchained, correct?

Without fast movement that's not going to work. It doesn't say 'the standard monk fast movement' or anything similar. I disagree that barbarian/bloodrager fast movement wouldn't work, but that's just 10' anyway. Bloodine arcana works with non-sorc spells for example.