This would not be an issue: "maybe they did not qualify for the first stimulus payment of $500 per dependent."

Because there is no Splitting of 2020. What matters is, on the 2020 form, the amounts you were entitled to or not (no splitting of 2020) against amounts paid in advance.

If the child qualifies and was born at any time in 2020, they are eligible for the $500 and the $600 as long as their other qualifications (filing status, income) are met. Just like anyone dying on Jan 1 or any date in 2020 is still considered eligible for the amount, as long as the other qualifications are met.

There is no splitting 2020. That tax return is the reconciliation of what they got, against both amounts that they qualify for. Not against each amount. You cannot qualify for only 1 of these, in 2020. Of course, you might have gotten only one of the amounts in advance.