We establish asymptotically Gaussian fluctuations for functionals of a large class of spin models and strongly correlated random point fields, achieving near-optimal rates. For spin models, we demonstrate Gaussian asymptotics for the magnetization for a wide class of ferromagnetic spin systems on Euclidean lattices, in particular those with continuous spins. Specific applications include, in particular, the celebrated XY and Heisenberg models under ferromagnetic conditions, and more broadly, systems with very general rotationally invariant spins in arbitrary dimensions. We address both the setting of free boundary conditions and a large class of ferromagnetic boundary conditions, and our CLTs are endowed with near-optimal rate. Our approach leverages the classical Lee-Yang theory for the zeros of partition functions, and subsumes as a special case results of Lebowitz, Ruelle, Pittel and Speer on CLTs in discrete statistical mechanical models for which we obtain sharper convergence rates. In a different direction, we obtain CLTs for linear statistics of a wide class of point processes known as $α$-determinantal point processes which interpolate between negatively and positively associated random point fields. We contribute a unified approach to CLTs in such models; significantly, our approach is able to analyse such processes in dimensions $\ge 3$, where structural alternatives such as connections to random matrix theory are not available. A key ingredient of our approach is a broad, quantitative extension of the classical Marcinkiewicz Theorem that holds under the limited condition that the characteristic function is non-vanishing only on a bounded disk. In spite of the general applicability of the results, our rates for the CLT match the classic Berry-Esseen bounds for independent sums up to a log factor.