Context. It is now well-known that pre-main-sequence models with inflated radii should be taken into account to simultaneously reproduce the colour-magnitude diagram and the lithium depletion pattern observed in young open star clusters. Aims. We tested a new set of pre-main-sequence models that include radius inflation due to the presence of starspots or to magnetic inhibition of convection. We used five clusters observed by the Gaia-ESO Survey that span the age range ∼10−100 Myr, in which these effects could be important. Methods. The Gaia-ESO Survey radial velocities were combined with astrometry from Gaia EDR3 to obtain clean lists of high-probability members for the five clusters. A Bayesian maximum likelihood method was adopted to fit the observed cluster sequences to theoretical predictions to derive the best model parameters and the cluster reddening and age. Models were calculated with different values of the mixing length parameter (α_ML = 2.0, 1.5 and 1.0) for the cases without spots or with effective spot coverage β_spot = 0.2 and 0.4. The models were also compared with the observed lithium depletion patterns. Results. To reproduce the colour-magnitude diagram and the observed lithium depletion pattern in Gamma Vel A and B and in 25 Ori, both a reduced convection efficiency, with α_ML = 1.0, and an effective surface spot coverage of about 20% are required. We obtained ages of 18_−4.0^+1.5 Myr and 21_−3.0^+3.5 Myr for Gamma Vel A and B, respectively, and 19_−7.0^+1.5 Myr for 25 Ori. However, a single isochrone is not sufficient to account for the lithium dispersion, and an increasing level of spot coverage as mass decreases seems to be required. On the other hand, the older clusters (NGC 2451 B at 30_−5.0^+3.0 Myr, NGC 2547 at 35_−4.0^+4.0 Myr, and NGC 2516 at 138₋₄₂⁺⁴⁸ Myr) are consistent with standard models (i.e. α_ML = 2.0 and no spots) except at low masses: a 20% spot coverage appears to reproduce the sequence of M-type stars better and might explain the observed spread in lithium abundances. Conclusions. The quality of Gaia-ESO data combined with Gaia allows us to gain important insights on pre-main-sequence evolution. Models including starspots can provide a consistent explanation of the cluster sequences and lithium abundances observed in young clusters, although a range of starspot coverage is required to fully reproduce the data.