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The Toad Birds in fact aren't birds, but just flying Toad ragdolls, smaller than the other members of that species. They seem to be made with the video game Garry's Mod.
Here is the detail of the appearances of the Toad Birds:
SM64: ṩṩἔᾗмὄḋᾗᾄʀ 7 (TOAST edition)
Besides being the first appearance of the Toad Birds in SMG4's videos, this episode is also their main appearance. They appear in the 11st skit, in which Roprinplup14 (Enzo) asks Mario to hold a bag for him. Mario asks what it is, and Enzo tells him it was "food for the Toad Birds", and quickly runs into another room of the castle. Mario yells after him that Toad Birds aren't a real thing, only to be attacked by an aggressive pack of flying Toad chanting "MINE! MINE! MINE! MINE! MINE!..." a few seconds later. That trick was presumed to be the revenge that Enzo wanted to get on Mario after this last one ruined his 2013 birthday party, but he in fact really took it only five months later, at his 2014 birthday.
In this other blooper, the Toad Birds reappear in the challenge called "The Toad Survival Island" by Kristi Ceka (the ideas of the challenges of this edition of the War of the Fat Italians are coming from some of SMG4's fans on YouTube). The "Toad Survival Island" is an Island full of Toad, and some Toad Birds are flying at the background, behind the other Toad.