Magnetic machine-learning potentials (MLPs) now reach near-first-principles accuracy on the spin-lattice potential energy surface, but the dynamics and sampling frameworks that convert this accuracy into quantitative finite-temperature thermodynamics have lagged behind. Landau-Lifshitz-Gilbert spin-lattice dynamics fixes the local moment magnitude and incurs O(N) MLP evaluations per integration step, while hybrid molecular-dynamics/Monte-Carlo lacks rigorous isothermal-isobaric sampling and remains expensive. We introduce TSPIN, which promotes the spin to a canonical pair (Sᵢ,πᵢ) alongside the lattice (Rᵢ,pᵢ) within a Nosé-Hoover-chain / Martyna-Tobias-Klein construction, delivering rigorous canonical and isothermal-isobaric sampling, native access to longitudinal spin fluctuations through an unconstrained spin amplitude, and one MLP evaluation per integration step. Applied to itinerant Co and localized multiferroic BiFeO₃, TSPIN matches the MD/MC reference thermodynamics of Co at substantially lower cost and reproduces the Curie and Néel temperatures within ∼ 7% and ∼ 2% of experiment, respectively. The same unconstrained-amplitude dynamics resolves contrasting spin-amplitude behavior: pronounced spin-modulus softening in Co, but a nearly temperature-independent high-spin Fe³⁺ moment in BiFeO₃. TSPIN thereby promotes magnetic MLPs from accurate energy models to predictive finite-temperature simulation engines.