Finite battery lifetime and low computing capability of size-constrained wireless devices (WDs) have been longstanding performance limitations of many low-power wireless networks, e.g., wireless sensor networks and Internet of Things. The recent development of radio frequency-based wireless power transfer (WPT) and mobile edge computing (MEC) technologies provide a promising solution to fully remove these limitations so as to achieve sustainable device operation and enhanced computational capability. In this paper, we consider a multi-user MEC network powered by the WPT, where each energy-harvesting WD follows a binary computation offloading policy, i.e., the data set of a task has to be executed as a whole either locally or remotely at the MEC server via task offloading. In particular, we are interested in maximizing the (weighted) sum computation rate of all the WDs in the network by jointly optimizing the individual computing mode selection (i.e., local computing or offloading) and the system transmission time allocation (on WPT and task offloading). The major difficulty lies in the combinatorial nature of the multi-user computing mode selection and its strong coupling with the transmission time allocation. To tackle this problem, we first consider a decoupled optimization, where we assume that the mode selection is given and propose a simple bi-section search algorithm to obtain the conditional optimal time allocation. On top of that, a coordinate descent method is devised to optimize the mode selection. The method is simple in implementation but may suffer from high computational complexity in a large-size network. To address this problem, we further propose a joint optimization method based on the alternating direction method of multipliers (ADMM) decomposition technique, which enjoys a much slower increase of computational complexity as the networks size increases. Extensive simulations show that both the proposed methods can efficiently achieve a near-optimal performance under various network setups, and significantly outperform the other representative benchmark methods considered.