I had a look at the video you linked and it is the doctored footage. Again, why would the studio allow such an obvious oversight to end up in the final production? This is a link to the 1955 re-release trailer (before digital technology) and you can see the same shot at the 1:25 timestamp with no hanging figure:

https://youtu.be/KkiGySEBCHY?si=h4mBvCnUmGut9tOV

As for exotic birds not appearing anywhere else, the crane can be seen at several points of the “If I Only Had a Heart” sequence around the Tin Man’s cabin, even spreading his wings at the 0:12 timestamp here:

https://youtu.be/u_6S1N5RZrk?si=0mp90Wlckc1xbzIB

The crane was brought in from the Los Angeles Zoo (probably with a zoo employee) to bring colour and dimension to the film as in the original novel Oz is described as being full of beautiful flowers and birds. The Wizard of Oz was a prestige film and while some goofs did end up in the Final Cut (eg: Dorothy not wearing the Ruby Slippers for a brief shot with the Scarecrow and the talking trees), such a glaring and unsettling visual would not have been left in. Filming was often stopped for minor mistakes so why not this? While I can understand being cautious about studios and companies as there are instances of films being altered, this is not the case of The Wizard of Oz, perhaps the most protected and studied film in history.

Hope this clears it up somewhat :)