Motivated by realistic hardware considerations of the pre-fault-tolerant era, we comprehensively study the impact of uncorrected noise on typical quantum circuits. We first show that any noise `truncates’ most quantum circuits to effectively logarithmic depth, in the task of estimating observable expectation values. We then prove that quantum circuits under any non-unital noise exhibit lack of barren plateaus for cost functions composed of local observables. But, we also design an efficient classical algorithm to estimate observable expectation values of typical quantum circuits within any target constant accuracy, in any circuit architecture. Taken together, our results showcase that, unless we carefully engineer the circuits to take advantage of the noise, it is unlikely that noisy quantum circuits provide any quantum advantage for algorithms that output observable expectation value estimates, like many variational quantum machine learning proposals.