  We study the quantum many-body instabilities of interacting electrons with
SU(2)$\times$SU(2) symmetry in spin and orbital degrees of freedom on the
triangular lattice near van-Hove filling. Our work is motivated by effective
models for the flat bands in hexagonal moir\'e heterostructures like twisted
bilayer boron nitride and trilayer graphene-boron nitride systems. We consider
an extended Hubbard model including onsite Hubbard and Hund's couplings, as
well as nearest-neighbor exchange interactions and analyze the different
ordering tendencies with the help of an unbiased functional renormalization
group approach. We find three classes of instabilities controlled by the
filling and bare interactions. For a nested Fermi surface at van-Hove filling,
Hund-like couplings induce a weak instability towards spin or orbital density
wave phases. An SU(4) exchange interaction moves the system towards a Chern
insulator, which is robust with respect to perturbations from Hund-like
interactions or deviations from perfect nesting. Further, in an extended range
of fillings and interactions, we find topological $d\pm id$ and
(spin-singlet)-(orbital-singlet) $f$-wave superconductivity.
