It really sounds like there is a communication barrier somewhere between the dance school and you, but you will know better than us whether that rests more with the school, more with you, or equally between the two.
Does your school offer a termly watch week or other way to see the children and what they are working on? Do they ever provide written feedback outside of the exam forms? If there’s absolutely none of that and you can’t sneak a look through a studio window or anything, then it’s worth thinking about this for the future. This is not how all schools work and strong communication as dancers progress makes for dancers with more awareness and responsibility for their own work as they get older.
Equally, emailing the school to ask politely about progress and what to work on does not make you neurotic, ‘that parent’, or a crazy dance mum. Every school runs a little differently and the only way to actually know what is happening is to stop speculating and address the situation with her teacher. My dancer has studied at three different schools over the years and each one of them would have a different strategy that would explain what you describe, just from the different set up for each. You need clarity from the person who knows. Just write concisely and without all the extra thoughts you’ve attached to this already - don’t try to couple it with a complaint. You need to know if any important exam dates need to be on your calendar and you need to check she is making an expected level of progress. Say she enjoys the lessons. Short and sweet.
If the response makes you twitchy or you don’t get a response over the course of a week, then you have the information you need to look at other options. All schools are not equal and it is okay to move around to find the right fit for your family! Nothing dramatic needed, perhaps even just quietly looking for something different for September. But it may actually be very clear from the response that all is well and there’s been a misunderstanding. So sort the communication first and then you are better prepared for finding out that DD is mixing up left and right or gets the giggles or whatever other reason the teacher may have thought it best to wait till the next exam session.
Remember schools often pride themselves on extremely high pass rates - and that comes from only entering those they are absolutely sure will
pass with flying colours.