I will be honest here (dance teacher). Most children take dance recreationally - for a few hours a week, which is fine as a form of fitness and a hobby.
The problem with dancing at a recreational level (a few hours a week) is that if the child gets to age 14 or so and decides that they want a career in dance, if at a high level (such as aiming for ballet school or one of the top conservatoires) then most of the time it's too late if they leave it until their teens to start training on a more serious level.
A dancer aiming for a career in the industry ideally needs to start training around age 7 or 8, to maximise their potential and to place them on an equal footing at auditions for full time schools (whether this is for entry at age 11, 14, 16 or 18). This is very young to make a career choice, but not unusual - you see young sports persons joining their county teams, swim and gymnastic squad training starts around this age or even younger etc. and of those who do opt for an intensive training many will drop out or change their mind along the way, which makes the whole process a bit of a gamble. This isn't to say that dance has no value as a recreational hobby as it is exercise and gives a creative outlet and much enjoyment can be had from dancing just for fun, but for a career in dance the earlier the training begins the better, as it isn't something that you can learn in a short space of time. Like any sport, you get out what you put in, and if your child is attending 3 hours of 'fun' classes a week they are not going to be improving at the same rate as a child of the same age who attends 12+ hours of classes a week in the core subjects (ballet, tap and modern) plus additional classes such as jazz, contemporary, acrobatics and conditioning/technique classes. To some degree, natural ability and potential also comes into it, but talent is of no use if it is not nurtured and trained correctly. And like in any sport or hobby that requires you to aim to be the best in your field you will sometimes come across pushy and ambitious parents and children.
And yes to ballet being the core to all forms of dance, even if your child has no intention of becoming a ballerina it is still an important core subject if they want to pursue dance as a career. Many performers now are aiming to become 'triple threat' performers (equally able in singing, acting and dance) as they are in high demand in the industry. It's fiercely competitive and places (and funding) for full time schools and colleges are in high demand. There are other dance/theatre related career options besides performing - I have former students who are now a theatre technician, performing arts teacher, fitness instructor, contemporary dance choreographer and one former student who runs a successful dance school of her own.
If a child wants to take dance as a career the cost of classes and extras (dancewear, costumes, exam fees etc) can be expensive, but this should not be a reason to avoid start dance classes as a hobby, as long as you are clear with yourself at the beginning what you want for your child and how much time and money that you are prepared to commit to it, then it should not be a problem. Issues only arise when parents want a recreational cost and level of commitment, but expect an intensive training level of progress. If you want your child to excel to a professional level then you have to invest into their training (again, just like in any sport). A supportive (but not pushy) parent (or parents) is the best asset a young dancer can have, as without the right support from home it is nearly impossible to pursue it on a serious level.
(Dance Parent = taxi driver, hairdresser, chaperone, seamstress, cheerleader, and personal assistant , to name a few. . . . ! )
This doesn't mean that if a child is a recreational dancer or doesn't seem to have a vast amount of natural ability that they should just give up. Dance classes are a great form of fitness and exercise, they encourage coordination, self discipline and confidence.
I see the dance classes that I teach as like a train journey - only a few of the students that I teach will have their eye on the terminus of the train track (a career in dance) as their final destination and many will disembark at stations along the way (starting secondary school, GCSE's, University) , but it doesn't mean that their journey is any less valuable than those who are in it for the long haul. It's about the skills that they learn that they will carry through with them into their adult life, the experiences had and the friendships made along the way.
The most important thing is that you find the right dance school for your child (and for your expectations as a parent) and that your child is happy and enjoying it. If at any point your child is no longer enjoying their dance classes then it's time to either find a new dance school where your child will thrive, or to let the dance classes go before it becomes something that they will look back on as an unhappy time in their life. Dance class should be a child's happy place, whether that's for one evening a week or every day, and no child should be forced to undertake any training that leaves them unhappy, exhausted or unable to cope.