You are the applicant. Your child cannot apply for a passport.
With regards to signing the actual passport, according to the website, if your child is 11yrs old or older, they are encouraged to sign it. However, even if they are younger and sign it, it's still valid:
The signature block on page 3 of your child’s travel document:
doesn’t need to be signed if your child is under 16
children between the ages of 11-15 are encouraged to sign it
if your child under the age of 11 signs it, it’s still valid
MUST be left blank if it isn’t signed by your child
However, if you or anyone other than your child signs their passport, it will be invalid. So please remember that no one can sign it other than your child.
I understand that it can be a bit confusing since the parent is the one submitting the application but they don't own the passport.
Thanks...I remember a customs agent somewhere along the line told us that as well, so we've been careful since. But it is confusing, and the logic is fuzzy to me.