We develop a self-contained theoretical framework that classifies the topological phases and critical phenomena of classical pyrochlore magnets with arbitrary spin $S$, subject to competing exchange and single-ion anisotropies. Working first in the monopole-free (ice-rule) limit: in the small-$w$ regime, where the single-ion term favors low spin amplitudes, dualities reveal a dichotomy: integer spins exhibit a continuous 3D $XY$ deconfinement transition, whereas half-integer spins remain in a $U(1)$ Coulomb liquid without any transition. In the large-$w$ regime, where the local spin amplitudes are maximized ($|S^z|=S$), the macroscopic flux is quantized to multiples of $2S$. By mapping the defect structure to topological loop gases, we prove that the compatibility between the physical ice rule and the modular ${\mathbb{Z}}_{2S}$ flux conservation holds at the vertex level if and only if $S\le 3/2$. For $S=3/2$, this maps the system to the 3-state Potts model, whose symmetry-allowed cubic invariant drives a first-order transition. For $S\ge 2$, no mapping to a discrete clock model exists. Using a decomposition of the partition function, we show that the only processes sensitive to the discrete ${\mathbb{Z}}_{2S}$ structure carry exponentially small statistical weight and correspond to irrelevant operators at the 3D $XY$ fixed point, so the deconfinement transition is of the 3D $XY$ type. Finally, incorporating thermal monopoles, we show that they act as a symmetry-breaking effective magnetic field that severs defect strings. Consequently, the continuous transitions are rounded into crossovers, whereas the first-order $S=3/2$ transition is predicted to survive at finite temperatures, terminating at a critical endpoint. Classical Monte Carlo simulations for $S$ up to $7/2$ corroborate the predicted crossover behavior at the accessible temperatures; these temperatures lie above the estimated critical endpoint, so the predicted survival of the $S=3/2$ first-order transition remains to be tested directly.