  We use the topological heavy fermion (THF) model and its Kondo Lattice (KL)
formulation to study the symmetric Kondo state in twisted bilayer graphene. Via
a large-N approximation, we find a symmetric Kondo (SK) state in KL mode at
fillings $\nu=0,\pm 1,\pm 2$. In the SK state, all symmetries are preserved and
the local moments are Kondo screened by the conduction electrons. At the
mean-field level of the THF model at $\nu=0,\pm 1, \pm 2, \pm 3$, we also find
a similar symmetric state. We study the stability of the symmetric state by
comparing its energy with the ordered states and find the ordered states to
have lower energy. However, moving away from integer fillings by doping holes
to the light bands, we find the energy difference is reduced, which suggests
the loss of ordering and a tendency towards Kondo screening. In order to
include many-body effects beyond the mean-field approximation, we perform
dynamical mean-field theory (DMFT) calculations on the THF model. We find the
spin susceptibility follows a Curie behavior at $\nu=0, \pm 1,\pm 2$ down to
$\sim 2\text{K}$ where the onset of screening of the local moment becomes
visible. This hints to very low Kondo temperatures at these fillings, in
agreement with the outcome of our mean-field calculations. At non-integer
filling $\nu=\pm 0.5,\pm 0.8,\pm 1.2$ DMFT shows deviations from a
$1/T$-susceptibility at much higher temperatures, suggesting a more effective
screening of local moments with doping. Finally, we study the effect of a
$C_{3z}$-rotational-symmetry-breaking strain via mean-field approaches and find
that a symmetric phase (that only breaks $C_{3z}$ symmetry) can be stabilized
at sufficiently large strain at $\nu=0,\pm 1, \pm 2$. Our results suggest that
a symmetric Kondo phase is strongly suppressed at integer fillings, but could
be stabilized either at non-integer fillings or by applying strain.
