To me, this "Archaeopteryx" almost exactly resembles something else, the fascinating historical phenomenon called Proavis.
Proavis, or Tetrapteryx as some four-winged interpretations were called, was a hypothetical prehistoric creature that was proposed in the early 20th century as a best guess at what the unknown ancestor of birds could have looked like. The illustration above was drawn in 1926 by Gerhard Heilmann, a Danish artist and amateur scientist who argued that birds evolved from non-dinosaurian archosaurs like Euparkeria. In his 1916 book Vor Nuvaerende Viden om Fuglenes Afstamning and the 1926 English translation The Origin of Birds, he presented Proavis as the imagined midpoint between a scaly ground-running archosaur and Archaeopteryx, which at the time held the title of The First Bird.
Other versions of the same hypothesis, like William Beebe's Tetrapteryx above, were published and discussed around the same time, but it was Heilmann's Proavis that gained immense popularity to the point that bird evolution was considered essentially "solved" for decades. It was also painted by Zdeněk Burian, one of the Old Greats of palaeoart, which kept the concept alive in dinosaur books for decades as well.
Of course further study has shown this hypothesis to be incorrect and that birds are instead members of Dinosauria (and honestly Heilmann either missed or ignored a lot of evidence for a dinosaurian origin of birds even in the 1910s), but the Proavis to me remains a beautiful and fascinating concept that represents scientists and artists striving to understand the prehistoric world and the passage of evolution, much like we still do today!
And of course, its popularity in the early 20th century put it at the perfect time for Fantasia's artists to take... let's say heavy inspiration from Heilmann's imaginary Proavis when depicting a creature that was intended to be Archaeopteryx the whole time! The pattern of feathers matches up almost exactly, although the larger leg wings might have been inspired by Beebe's Tetrapteryx as well:
So to get back to your original question that led to this whole deep dive, artists didn't actually Draw Archaeopteryx Like That except when they were mistakenly drawing something that wasn't Archaeopteryx at all! If you want to read more about the Proavis and Tetrapteryx I recommend this Tetrapod Zoology blog post by Darren Naish, he does into more depth about the history of the concept and some of the unusual evolutionary ideas that Heilmann used to arrive at this weird and cool imaginary creature!