  This paper provides a self-contained exposition of coorbit spaces associated
to integrable group representations and quasi-Banach function spaces, and at
the same time extends and simplifies previous work. The main results provide an
extension of the theory in [Studia Math., 180(3):237-253, 2007] from groups
admitting a compact, conjugation-invariant unit neighborhood to arbitrary
(possibly nonunimodular) locally compact groups. In addition, the present paper
establishes the existence of molecular dual frames and Riesz sequences as in
[J. Funct. Anal., 280(10):56, 2021] for the full scale of quasi-Banach function
spaces. The theory is developed for possibly projective and reducible unitary
representations in order to be easily applicable to well-studied function
spaces not satisfying the classical assumptions of coorbit theory. Compared to
the existing literature on quasi-Banach coorbit spaces, all our results apply
under significantly weaker integrability conditions on the analyzing vectors,
which allows for obtaining sharp results in concrete settings
