The reuse of animation in this way is attributed to director/producer Wolfgang Reitherman, and was reportedly done because the animation had already proved to work well in its previous use.
From The Real Truth About Disney’s “Recycled Animation” (Medium crosspost), as reported by Jim MacQuarrie regarding an interview with Floyd Norman about a similar viral video (emphasis mine):
He nodded, laughed and said, “That was Woolie Reitherman.” A quick check of Wolfgang Reitherman’s IMDB page confirms that nearly every Disney film shown using recycled footage in these videos is one he directed, most notably the Jungle Book, Robin Hood, the Sword in the Stone, Winnie the Pooh, 101 Dalmatians and the AristoCats.
“It’s actually harder and takes longer to redraw an existing sequence,” Norman told me, “it’s a lot faster and easier to just do new animation, and it’s a lot more fun for the animators. But Woolie liked to play it safe and use stuff he knew would work. That’s all it was.”
Milt Kahl, another of Disney's "Nine Old Men" among Norman and Reitherman, confirms in a 1976 interview with Michael Barrier and Milton Gray:
Barrier: I can't see how the re-use of animation saves money.
Kahl: It doesn't. The funny thing is that most of the time they spend more money trying to figure out how to re-use it than they would [animating the scene from scratch]. I'd rather see it animated from scratch; even if it's a little bit amateurish, at least it's fresh and new.
This primary-source reporting was then cited in the Wikipedia article on Wolfgang Reitherman, as notes 15 and 18 respectively:
During his tenure, he [Wolfgang Reitherman] frequently used "recycled" or limited animation from prior works, presumably because it was a safer method for a quality product, though it was in fact more labor-intensive15, not because it was supposedly cheaper.16,17 Reitherman's use of recycling animation proved to be controversial within the studio as animator Milt Kahl lamented its use stating "I detest the use of—it just breaks my heart to see animation from Snow White used in The Rescuers. It kills me, and it just embarrasses me to tears."18
It is possible that Reitherman's original motivation was saving money or tightening deadlines, and that the animators took a cynical or futile view, but in any case the practice is associated specifically with Reitherman and in available interviews fellow animators felt it was less about money than about using what had worked in previous films.
For what it's worth, MacQuarrie's postscript update cites different reasons for Beauty and the Beast, in a conversation with Gary Trousdale that still cites Reitherman's precedent:
“The scene from Beauty and the Beast that we re-used was done for time, but not money. (well…time IS money, but that’s another story). We were just days from our final deadline to deliver, and we had an entire dance sequence (the last scene of the movie, not the ballroom) to do. Everyone was booked and busy, and we knew damn well that Woolie had established this precedent, so we took the Sleeping Beauty dance, re-sized and re-positioned it, and gave the note ‘Note to Clean-up: clean up Aurora as Belle, clean up Prince Charming as Beast.’”