  Flat bands in magic-angle twisted bilayer graphene serve as the arena where
exotic correlation physics unfolds. Previous studies have shown that phonons,
through mediating an effective electron-electron interaction, can play a
crucial role in determining various electronic phases. In this study, we
project the full electron-phonon coupling vertex, derived from microscopic
tight-binding lattice calculations, onto the basis of the topological heavy
fermion model [Song and Bernevig, PRL 129, 047601 (2022)], and identify the
significance of each phonon mode. With phonon fields integrated out, an on-site
anti-Hund's interaction $\hat{H}_{\rm A}$ on the moir\'e-scale local
$f$-orbitals is obtained, with strengths 1 to 4 meV. We solve the
phonon-induced multiplet splittings, and then elaborate on phonon-favored
symmetry-breaking orders at even-integer fillings. Through systematic
self-consistent Hartree-Fock calculations, we uncover a tight competition
between $\Gamma$-phonon-favored orbital orders, $K$-phonon-favored inter-valley
coherent orders, and the kinetic and Coulomb-favored orders. A Hund's
interaction $\hat{H}_{\rm H}$ with strengths 1 to 3 meV, which originates from
the carbon atom Hubbard, and partly counteracts the effect of $\hat{H}_{\rm
A}$, is also discussed. In the end, we explore the possibility of finding an
exotic Dirac semi-metal at the charge-neutrality point which is formed solely
by $c$-electrons, while $f$-impurities exhibit a symmetric Mott gap by forming
non-degenerate singlets under $\hat{H}_{\rm A,H}$. Experimental features that
distinguish such a state are discussed.
