A growing disquiet has emerged in recent years that standard stellar models are at odds with observations of the colour-magnitude diagrams (CMDs) and lithium depletion patterns of pre-main sequence stars in clusters. In this work we select 1246 high probability K/M-type constituent members of five young open clusters (5-125 Myr) in the Gaia-ESO Survey to test a series of models that use standard input physics and others that incorporate surface magnetic fields or cool starspots. We find that: standard models provide systematically under- luminous isochrones for low-mass stars in the CMD and fail to predict Li- depletion of the right strength at the right colour; magnetic models provide better CMD fits with isochrones that are ~1.5-2 times older, and provide better matches to Li depletion patterns. We investigate how rotation periods, most of which are determined here for the first time from Transiting Exoplanet Survey Satellite data, correlate with CMD position and Li. Among the K-stars in the older clusters we find the brightest and least Li-depleted are the fastest rotators, demonstrating the classic 'Li-rotation connection' for the first time at ~35 Myr in NGC 2547, and finding some evidence that it exists in the early M-stars of NGC 2264 at < 10 Myr. However, the wide dispersion in Li depletion observed in fully convective M-dwarfs in the {gamma} Vel cluster at ~20 Myr appears not to be correlated with rotation and is challenging to explain without a very large (>10 Myr) age spread.