Drones have the potential to revolutionize power line inspection by increasing productivity, reducing inspection time, improving data quality, and eliminating the risks for human operators. Current state-of-the-art systems for power line inspection have two shortcomings: (i) control is decoupled from perception and needs accurate information about the location of the power lines and masts; (ii) obstacle avoidance is decoupled from the power line tracking, which results in poor tracking in the vicinity of the power masts, and, consequently, in decreased data quality for visual inspection. In this work, we propose a model predictive controller (MPC) that overcomes these limitations by tightly coupling perception and action. Our controller generates commands that maximize the visibility of the power lines while, at the same time, safely avoiding the power masts. For power line detection, we propose a lightweight learning-based detector that is trained only on synthetic data and is able to transfer zero-shot to real-world power line images. We validate our system in simulation and real-world experiments on a mock-up power line infrastructure. We release our code and datasets to the public.