I think it helps to look at the wider picture here:
Firstly, learning to read music and to play a musical instrument has been shown to be beneficial to other studies; which is one of the reasons that public schools are so keen to encourage musical studies. (Likewise sport.)
Secondly; learning music, and especially playing in a band or an orchestra helps children to develop character, to socialise and to learn social skills.
Thirdly; music provides a safe and creative outlet for the talents, energy and frustrations of young people, and also helps them to avoid other less desirable distractions and temptations in their teenage years. Frankly I would be quite happy if our children didn't play another note when they reached adulthood as long as they got there safely.
I think we also need to remember that children's interests tend to be transient, and are subject to the inevitable strains of growing up, peer pressure and other interests; so a lack of interest in music is often temporary. Pushing children too hard to do something they may not want to do for a short while is usually counterproductive in my experience!
However, musical instruments are usually chosen for children for entirely practical reasons; such as cost, size, convenience; but that instrument does not necessarily enthuse the youngster in the ways that we might hope. As an example, our own son first played on a piano at church when he was about three years old. He was clearly fascinated by the instrument, climbed onto the stool, and kneeling he started to play handfulls of notes in typical three year old fashion! I showed him how to play simple chords, which obviously pleased him, and before long he was able to make musical sounds of his own.
Having seen this interest we gave him a Yamaha electronic keyboard for Christmas, which he loved, and then progressed to a piano. However, whilst he would spend hours improvising and playing tunes by ear (with remarkable accuracy), he just would not settle to lessons, and his sight reading remained poor. Then, at the age of 13 on a cold, wet Sunday afternoon he got to play on a real pipe organ for the first time. It would not be an exaggeration to say that this experience quite literally changed his life, as he immediately saw a reason to learn to play properly, and is now quite obsessive about playing the instrument. Unfortunately, Mrs Swellbox and our daughter do not share our obsession, so our problems tend to be the reverse of those discussed in this thread!
However, I think this story does perhaps demonstrate that it is sometimes a matter of finding the right instrument, rather than just plugging away at whatever happens to be available.
Ironically, our piano is used more now than ever; not to play piano music, but to practice hymns and organ music in comfort!