  We investigate the nature of the ground-state of the spin-$\frac{1}{2}$
Heisenberg antiferromagnet on the $shuriken$ lattice by complementary
state-of-the-art numerical techniques, such as variational Monte Carlo (VMC)
with versatile Gutzwiller-projected Jastrow wave functions, unconstrained
multi-variable variational Monte Carlo (mVMC), and pseudo-fermion/Majorana
functional renormalization group (PF/PM-FRG) methods. We establish the presence
of a quantum paramagnetic ground state and investigate its nature, by
classifying symmetric and chiral quantum spin liquids, and inspecting their
instabilities towards competing valence-bond-crystal (VBC) orders. Our VMC
analysis reveals that a VBC with a pinwheel structure emerges as the
lowest-energy variational ground state, and it is obtained as an instability of
the U(1) Dirac spin liquid. Analogous conclusions are drawn from mVMC
calculations employing accurate BCS pairing states supplemented by symmetry
projectors, which confirm the presence of pinwheel VBC order by a thorough
analysis of dimer-dimer correlation functions. Our work highlights the
nontrivial role of quantum fluctuations via the Gutzwiller projector in
resolving the subtle interplay between competing orders.
