First off, I think the other answer is excellent.
You seem to have two questions, so I’ll answer them separately:
1p, 2f, 3g orbitals. In my attempt to reconcile this, I was thinking perhaps these are just so-called additional basis functions [...] to better approximate the electron behaviour
Yes, exactly. Technically speaking the s, p, d, f functions are only accurate for the one-electron hydrogen atom anyway. Moreover, we usually use Gaussian functions for most basis sets, so we’re just attempting to approximate the real electron density in a multi-electron system.
As you said, we use higher angular momentum basis functions (like 1p, 2f, 3g, h, i, etc.) to better approximate the angular components (polarization) of the electron density. Sometimes we use higher principal quantum numbers to better approximate the diffuse properties of the electron density (e.g., 2s, 3s, 4s for a hydrogen atom)
It’s important to understand that most depictions of orbitals or electron density are given as surfaces. Here’s a better picture - the orbital density of a tetrathiophene molecule.
Does this mean that there are only 1-electron orbitals in theories like DFT and HF, and no 2-electron orbitals.
Mostly, yes. In typical DFT and HF, we use 1-electron orbitals. We force double occupation in theories like RHF and ROHF, but the orbitals are solutions to 1-electron equations in both HF and traditional DFT.
There are electronic structure methods that use 2-electron orbitals. These geminal theories work over 2-electron orbitals. They’re not widely used (yet) but offer some advantages for certain properties like bond-breaking.
As an example, Assessing the Accuracy of New Geminal-Based Approaches, J. Phys. Chem. A, 2014, 118 (39), pp 9058–9068.
Our study indicates that these new geminal-based approaches provide a cheap, robust, and accurate alternative for the description of bond-breaking processes in closed-shell systems requiring only mean-field-like computational cost. In particular, the spectroscopic constants obtained from OO-AP1roG are in very good agreement with reference theoretical and experimental data.