It's not about the naming systems. It's about the approaches to

learning new pieces. Many solfege teachers are from Asia, where reading skill is often not prioritized. Learning new music often heavily relies on listening to recordings and copying demonstrations of the teacher. Those are easier for younger students than reading notes by themselves. However, the modern piano pedagogy emphasizes reading skill and view it as one of the most important fundamentals for long term development. It sounds like your daughter is caught in between the two philosophies. Plus,reading for her age is hard anyway. It may take her a while to adjust.

Again, what system was the first one?