Quantum Amplitude Estimation promises to accelerate Monte Carlo integration by replacing the classical ε∝ N^-1/2 sampling law with the Heisenberg-limited ε∝ N^-1 scaling, but whether any part of that advantage survives on present-day hardware is an empirical question. We report a systematic experimental benchmark of a Maximum-Likelihood Amplitude Estimation QMC pipeline on IonQ trapped-ion processors, comprising 60 hardware trials on IonQ Forte-1 across five settings(5 9 qubits, 100-200 shots per depth, linear exponential amplification schedules,  113 device-hours), together with 100 trials on IonQ Aria-1 Forte-1 noise models at 5, 9 19 qubits. The pipeline combines controlled-R_y rotational state preparation, Grover-based amplitude amplification and maximum-likelihood inference and is benchmarked against b_max^-1∫_0^b_maxsin^2x dx with b_max=π/5 ,given results emerge. First, the noiseless limit it attains ε∝ N^-0.88, close to the Heisenberg-scaling reference, reducing the error 54-74x over a 133x increase in oracle calls. Second, scaling does not survive on hardware. Every noisy backend saturates at an error floor of (2-5)x 10^-2, with fitted exponents of 0.04-0.15 on Forte-1 linear schedules and α≤0 on the vendor noise models: beyond the optimal shallow depth, however, deeper amplification provides no sustained accuracy gain and instead returns to the noise floor. Forte-1 hardware also outperformed IonQ's own Aria-1 noise model at every depth m ≥ 2, indicating that vendor noise models are bad predictors of MLAE accuracy. The practical implication for UQ is that on current devices the useful operating point is a shallow, schedule-tuned amplification depth chosen so that the deepest circuit lands near p 1/2 not the deepest schedule the coherence budget allows.