The Democrats are a liberal party; they believe in removing the legal and practical barriers to legitimate individual choices. Some families would like to have more children but are prevented from doing so by soaring costs. Hence, the need for a larger child tax credit.
The GOP, on the other hand, is split. The majority consists of right-wing liberals, who support the right of women to choose their number of offspring but put a higher priority on limiting the size and cost of government than on increasing the size of families. The minority, represented most conspicuously by J.D. Vance, sees tariffs and deportations as the principal means by which the Godly Society can be recreated. More children will be the inevitable result of the Godly Society. The child tax credit, while somewhat helpful, is at best a distraction in their eyes; it may do some good in the short run, but it props up the unholy status quo, which tolerates the illegitimate choice of limiting family size, in the long run.
If you were wondering why a party that purports to be “pro-family” opposed the tax credit, look no further than that.