I recently realised that the STO-nG basis functions were fit to 1s/20/3d STOs of various exponents, instead of being fit to STOs of higher principal quantum numbers (such as 2s/3p orbitals etc.).
If I understand what you are saying, this is incorrect (although feel free to clarify if I am not understanding the question).
Here is the paper documenting STO-nG for Na-Ar: https://doi.org/10.1063/1.1673374. The first sentence of the abstract:
"Least‐squares representations of the 3s and 3p Slater‐type atomic orbitals by a small number of Gaussian functions are presented."
However, it is true that they are expanded in terms of 1s and 2p Gaussians, which leads to your next question
Are there any GTO integration schemes for Cartesian GTOs of higher principal quantum numbers?
I am not aware of any codes that take advantage of higher principal quantum numbers. There likely isn't any need to.
The Molecular Electronic Structure Theory book (Helgaker et al), has this to say about STO-3G (page 290-291):
"Note the ... absence of a radial node in the STO-3G $2s$ orbital. In calculations using this basis, the $2s$ radial node will arise from a linear combination of the STO-3G 1s and 2s orbitals. The absence of a node in the STO-3G orbital is therefore not a problem."
These nodes arise from higher principal quantum numbers. Note that (almost) all basis set formats do not include principal quantum numbers. (A few do, but they are materials-oriented).