  It is a matter of current debate whether the gate-tunable superconductivity
in twisted bilayer graphene is phonon-mediated or arises from electron-electron
interactions. The recent observation of the strong coupling of electrons to
so-called $K$-phonon modes in angle-resolved photoemission spectroscopy
experiments has resuscitated early proposals that $K$-phonons drive
superconductivity. We show that the bandwidth-enhancing effect of interactions
drastically weakens both the intrinsic susceptibility towards pairing as well
as the screening of Coulomb repulsion that is essential for the phonon
attraction to dominate at low temperature. This rules out purely
$K$-phonon-mediated superconductivity with the observed transition temperature
of $\sim 1$ K. We conclude that the unflattening of bands by Coulomb
interactions challenges any purely phonon-driven pairing mechanism, and must be
addressed by a successful theory of superconductivity in moir\'e graphene.
