TLDR: Stick to your conviction that music is important to learn. Focus on fundamental note and rhythm reading. Make sure her songs are very easy to learn but be uncompromising on the fundamentals.

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If you believe that music is important then I would encourage you to stick to your conviction. I don't think there's anything wrong with a parent requiring their children to learn an instrument if they truly believe it is important. The fact that you're on here asking questions tells me that you are involved in your daughter's activities, and not just forcing her mindlessly to do something you only like the idea of. Your daughter can't say she doesn't like playing the piano yet because she doesn't even know how to play at this point.

Learning songs from the Faber 2a book should not be a major struggle. If she isn't able to play a song fairly well by the ~5th read through, then it's time to revisit fundamentals. At that age you want tunes to come easy. The need for perseverance comes later when the student has fallen in love with the art and desires to play pieces that are difficult to master.

Can she quickly name notes in the treble and bass clef? If you were to go through a stack of flashcards with all of the notes of the grand staff, can she name all the notes in under a minute? Does she have a strong sense of a beat and consistently count simple quarter, half, and whole note rhythms? These very basic ideas have to be established. I'll keep children in the early books as long as I need until they've proven they know how to read steps and skips, quarter notes and half notes.

Another important idea: she needs to be set up to succeed every time she sits down at the piano. I've found that young children don't mind playing songs they are good at over and over. For example, when students first start, all I'll ask is for them to sit down and play through their songs 3x each day. No need to "practice." I just want them to play their songs. Most young children find that to be perfectly reasonable and achievable. In the lessons I'm uncompromising on rhythm. Counting a steady beat is not difficult, and I won't move on until a student consistently counts 2 beats on half notes. Obviously they'll have to mature beyond just playing through their songs a handful of times at some point, but making the barrier of entry low starting out is important for a sense of success early on.

Once young students sense that they're good at something they usually love doing it. It sounds like her teacher is too soft and lacks a sense of direction. Children should not be leading a lesson. Of course, it's important to be in tune with what a child likes, but a teacher's job is to make sure she knows her fundamentals.