  Pinch point singularities, associated with flat band magnetic excitations,
are tell-tale signatures of Coulomb spin liquids. While their properties in the
presence of quantum fluctuations have been widely studied, the fate of the
complementary non-analytic features -- shaped as half-moons and stars --
arising from adjacent shallow dispersive bands has remained unexplored. Here,
we address this question for the spin $S=1/2$ Heisenberg antiferromagnet on the
kagome lattice with second and third neighbor couplings, which allows one to
tune the classical ground state from flat bands to being governed by shallow
dispersive bands for intermediate coupling strengths. Employing the
complementary strengths of variational Monte Carlo, pseudo-fermion functional
renormalization group, and density-matrix renormalization group, we establish
the quantum phase diagram. The U(1) Dirac spin liquid ground state of the
nearest-neighbor antiferromagnet remains remarkably robust till intermediate
coupling strengths when it transitions into a pinwheel valence bond crystal
displaying signatures of half-moons in its structure factor. Our work thus
identifies a microscopic setting that realizes one of the proximate orders of
the Dirac spin liquid identified in a recent work [Song, Wang, Vishwanath, He,
Nat. Commun. 10, 4254 (2019)]. For larger couplings, we obtain a collinear
magnetically ordered ground state characterized by star-like patterns.
