  We use the functional renormalization group equation for the effective
average action to study the non-Gaussian renormalization group fixed points
(NGFPs) arising within the framework of f(R)-gravity minimally coupled to an
arbitrary number of scalar, Dirac, and vector fields. Based on this setting we
provide comprehensible estimates which gravity-matter systems give rise to
NGFPs suitable for rendering the theory asymptotically safe. The analysis
employs an exponential split of the metric fluctuations and retains a
7-parameter family of coarse-graining operators allowing the inclusion of
non-trivial endomorphisms in the regularization procedure. For vanishing
endomorphisms, it is established that gravity coupled to the matter content of
the standard model of particle physics and many beyond the standard model
extensions exhibit NGFPs whose properties are strikingly similar to the case of
pure gravity: there are two UV-relevant directions and the position and
critical exponents converge rapidly when higher powers of the scalar curvature
are included. Conversely, none of the phenomenologically interesting
gravity-matter systems exhibits a stable NGFP when a Type II coarse graining
operator is employed. Our analysis resolves this tension by demonstrating that
the NGFPs seen in the two settings belong to different universality classes.
