"Do X or I'll quit" is a childish way to negotiate, its akin to a child throwing a tantrum if they don't get their way. It's extremely unprofessional and not something a team player would do. I think its clear why they took her up on her offer.
I disagree with this. It's really the only way to negotiate, pretty common. The wording is important here. If you word it as "Do X or I'll quit" it sounds childish, if you word it as "I believe the company should X, otherwise I can't see myself working here comfortably and will have to consider my resignation" it sounds professional, but the idea is the exact same.
Yes, in that they are both childish. Your version just added more weasel words. The second you bring up the "R" word as a threat you have played your hand; everything else is window dressing.
As I point out in another comment[1], not only is this not true, negotiation experts universally criticize this as a way to negotiate.