  Twisted $N$-layer graphene (TNG) moir\'e structures have recently been shown
to exhibit robust superconductivity similar to twisted bilayer graphene (TBG).
In particular for $N=4$ and $N=5$, the phase diagram features a superconducting
pocket that extends beyond the nominal full filling of the flat band. These
observations are seemingly at odds with the canonical understanding of the
low-energy theory of TNG, wherein the TNG Hamiltonian consists of one flat-band
sector and accompanying dispersive bands. Using a self-consistent Hartee-Fock
treatment, we explain how the phenomenology of TNG can be understood through an
interplay of in-plane Hartree and inhomogeneous layer potentials, which cause a
reshuffling of electronic bands. We extend our understanding beyond the case of
N = 5 realized in experiment so far. We decribe how the Hartree and layer
potentials control the phase diagram for devices with N $>$ 5 and tend to
preclude exchange-driven correlated phenomena in this limit. To circumvent
these electrostatic constraints, we propose a new flat-band paradigm that could
be realized in large-N devices by taking advantage of two nearly flat sectors
acting together to enhance the importance of exchange effects.
