An interesting piece of history showing what early judo looked like, though not having much to do with Yamashita. You can find some facts about captain Smith here:
Also, a short story with few photos here:
However, I don't think this was a unified approach to hand-to-hand combat in US Army of the day. As far as it is known Smith was instructor at Fort Myer and Fort Benning. The latter was used for basic training and considering the background it's probably where they filmed all that. Fort Benning was established in October 1918, so the film was probably from that era too. That's supported by an interesting detail. Those more familiar with military equipment will recognize that the "American" soldier in the drill wears the experimental helmet Model 5. Those were produced in small numbers (cca 2000). However, despite their excellent design, they were discarded by commanders who thought they too much resembled Germans' M16s (worn by the "German" soldier in the film). Only few were ever used in the field at the very end of war, making the unused pieces more suitable for training (while the "good stuff" was being sent to trenches).
In terms of actual techniques, the film clearly shows the type of judo/jujutsu Smith was exposed to while in Japan i.e. more combat oriented and probably not so strictly Kodokan.
For further reading his most known book "The Secrets of Jujitsu" published in 1920 can be found here:
I hope this helps.