Do British citizens (including British children born to EU citizens) need to apply for the EU Settlement Scheme?

No.  British citizens do not need to apply for pre-settled or settled status.

For information on whether your child may be British, we recommend reading the information on the PRCBC website (https://prcbc.org/our-work/),in particular this useful infographic: https://prcbc.files.wordpress.com/2019/03/children-and-their-rights-to-british-citizenship-march-2019.pdf.