Very interesting video. 'Unsoku undo' is quite similar to what I was taught, but the range of movements was more extensive... I guess it boils down to the school's emphasis on kata... although the Aikikai schools I've visited do not seem to focus on empty-hand kata...
I'm not sure if I'd designate the stuff we do in Kihon kozo (Basic Practices?) as kata per se. We do them as exercises to build the base fundamentals of motion for Aikido waza and for correct movement and posture in a kata and randori environment. All of the base movements found in Unsoku and Tegatana dousa are expanded upon to create the movements commonly seen when executing tai sabaki, waza and other movements of Aikido.
All the kihon kozo exercises build on each other, Unsoku (footwork) is the foundation for the next exercise, tegatana dousa (handblade movements e.g. strikes, thrusts, turns and twists) while using proper footwork, which then becomes the foundation for tegatana awase (moving ma ai, metsuke and footwork practice using tegatana) which then leads to sei chu sen no bogyo (metsuke, attack, reaction and entering exercise using tegatana and footwork) and onwards to the other kihon exercises.
Kata is a preset movement routine with a preset form, so in that light I'd say that Unsoku and Tegatana Dousa are kata, but the exercises that they operate as the foundation for are not preset, since they are designed to build reactions based on the unknown. There is no "focus on empty handed kata" however as you put it. Our fundamental kata, the Randori no kata is really based on a sword but is executed using tegatana or tanto.
I hope this clarifies.