According to Wikipedia, there are few models (?) for ∞-groupoids such as Kan complexes, Globular sets.
Instead of this this direction, most of the literature is about simplicial sheaves/stacks and their model structures.
Fibrant objects in the local projective model structure on simplicial presheaves
are precisely presheaves valued in Kan complexes that satisfy the descent (gluing) condition, so I am not sure what “instead” refers to,
since Kan complexes are one of the (geometric) models for ∞-groupoids.
Model structures are already present when you work with stacks in groupoids.
The reason why they are not mentioned explicitly that often
is because all objects are cofibrant and fibrant in the local model structure
on Grothendieck fibrations in groupoids.
Thus, one has no need to derive the internal hom functor, for example.
However, if one were to compute the limit or colimit of a diagram
of stacks in groupoids, the correct notion is not the strict (co)limit,
but rather the (co)limit with equalities replaced by isomorphisms
at appropriate places.
This is precisely the homotopy (co)limit that the machinery of model categories computes.
Model categories appear in the world of simplicial presheaves
for exactly the same reason they appear in the world of stacks
in groupoids modeled via Grothendieck fibrations:
there are maps (weak equivalences, or Morita equivalences
in the specific case of stacks in groupoids) that one wants to be invertible up to a homotopy,
but which are not invertible in any conventional sense.
Hence the need to use the machinery of model categories
(more generally, relative categories), which is designed
to treat precisely this issue.
