I am confused about formula 15.6.4. It does not seem to scale properly with the number of data points. For reference, the formula is

delta a_0 = sqrt(Delta Chi_nu^2)*sqrt(C_00)

Delta Chi_nu^2 is a chosen constant, and the matrix C is the inverse of the matrix alpha, which is a sum over the data points. The more data points, the larger alpha should be and the smaller C should be. This would be appropriate if we identify sqrt(C_00) as the uncertainty in the mean of the parameter of interest, but not the standard deviation of that parameter's distribution. The confidence interval delta a_0, on the other hand, should not get smaller as the number of data points grows.

What am I missing?

Why should delta a_0 not get smaller as the number of data points increases? a_0 is an estimate, derived from the data, of one of the parameters. The more data, the more reliable the estimate.