Doesn't a chiled under 24 who is a full time student living at home qualify for the full child tax credit?

No.

The full child tax credit of $2,000 is only for qualified children under the age of 17.

Please read this TurboTax article for more information.

Q. Doesn't a child under 24 who is a full time student living at home qualify for the full child tax credit?

A. No.  It's a common misunderstanding of the rules.  It just means the child can still be a dependent, no matter how much money he makes.  He qualifies for the $500 "Other dependent credit", instead. The child tax credit stops at age 17. 

There are two types of dependents, "Qualifying Children"(QC) and Other ("Qualifying Relative" in IRS parlance even though they don't have to actually be related). There is no income limit for a QC but there is an age limit, student status, a relationship test and residence test.

A child of a taxpayer can still be a “Qualifying Child” (QC) dependent, regardless of his/her income, if:

1. He is under age 19, or under 24 if a full time student for at least 5 months of the year, or is totally & permanently disabled
2. He did not provide more than 1/2 his own support. Scholarships are excluded from the support calculation
3. He lived with the parent (including temporary absences such as away at school) for more than half the year