I have not found strong effects of musical training on non-music related skills in child development in the published research. Intervention studies that are well-controlled, for example, with music training tested against other active training, single blind, are relatively few in number. The results are somewhat predictable. The music-related skills improve (obviously), while other changes may or may not be of importance to most people. See, for example:
Habibi A, Damasio A, Ilari B, Elliott Sachs M, Damasio H. (2018) Ann N Y Acad Sci. doi: 10.1111/nyas.13606.
Music training and child development: a review of recent findings from a longitudinal study:
https://www.ncbi.nlm.nih.gov/pubmed/29508399, https://dornsife.usc.edu/assets/sites/1247/docs/Music_training_and_child_development__a_review_of_recent_Annals_of_the_New_York_Academy_of_Sciences_Habibi_et_al-2018.pdf
Contrary to their better performance on musical tasks, in the
nonmusical cognitive tasks, children in the music group did not
perform better or show more notable improvement than those in both
control groups. We also did not observe any differences in cortical
thickness or cortical volume between the music and control groups,
outside of the auditory regions (e.g., in the inferior frontal gyrus).
However, children with music training showed significantly greater
neural activity during a color–word Stroop task in a network of brain
regions that are known to be involved in response inhibition.
Interestingly, this contrasts sharply with the practice of many prominent teachers, such as Rafe Esquith, who include music training into the curriculum: https://www.arts.gov/honors/medals/rafe-esquith
[His program for the fifth grade students] includes studying a musical instrument and one of Shakespeare's
plays for the entire year.
In view of this, I think that further studies are needed to study the effects of music training on development.
Finally, this question (or the studies such as above) do not address a very common reason for learning to play a musical instrument - just for the pleasure of it and for the beauty of making music.
SEE MORE:
Rafe Esquith: The Journey Is Everything (2007): https://www.giarts.org/article/journey-everything
I soon learned a basic truth: Students involved in arts education are
learning about things far beyond the art they study. When a child goes
off to play in an orchestra, he is not only learning to play the
violin or clarinet. He is also learning about discipline,
responsibility, teamwork, sacrifice, practice, correcting mistakes,
listening, and time management. That's not a bad set of skills for a
kid to have in his pocket. And to learn them and have fun at the same
time is a pretty neat trick.
Rafe Esquith: "Lighting Their Fires: Raising Extraordinary Children in a Mixed-up, Muddled-up, Shook-up World" (2009) Viking Penguin, New York, pp.29-30: https://www.amazon.com/Lighting-Their-Fires-Extraordinary-Muddled-up/dp/0670021083/
The world would be a better place if playing music was a requirement
for every child on the planet. Music is crucial for a child’s complete
development no matter his aptitude. For many reasons, kids who play
music have a better understanding of time than those who do not.
Einstein wrote about the connection between music and math, and
students who perform brilliantly in music often develop a love of
numbers. By necessity, musicians have a better understanding of time
because of the intricate meters and time signatures that keep a band
or orchestra together.
The benefits aren’t limited to the playing
itself. When a child plays music, he learns to be on time for
rehearsals. He also needs to plan his day and set aside time to
practice his instrument. For better or for worse, a student sees the
results of a lack of practice due to poor time management. Young
musicians learn not only to keep time on their own but to keep time
with others when playing in a group. All of the children with me at
the ball game play music, and all are always on time both in class and
for our extracurricular activities. It’s not a coincidence. Whether
you are able to give your child private lessons or make sure he plays
in school, music must not be optional. Reading music is as important
as reading itself.
Many schools today see it differently, and offer
music only as an elective. This is a mistake. Music training should be
required as part of a complete education. If your child reads music,
plays music, and makes it a part of his daily existence, his life will
be better for it. Through music, a student will learn to literally
play with others, and without realizing it will develop time
management skills that will be useful throughout the day, even when
the instrument is not being played.
