Aims. We present the analysis of a large sample of gamma-ray burst (GRB) X-ray light curves in the rest frame to characterise their intrinsic properties in the context of different theoretical scenarios. Methods. We determine the morphology, timescales, and energetics of 64 long GRBs observed by Swift/XRT without flaring activity. We furthermore provide a one-to-one comparison to the properties of GRBs with X-ray flares. Results. We find that the steep decay morphology and its connection with X-ray flares favour a scenario in which a central engine origin. We show that this scenario can also account for the shallow decay phase, provided that the GRB progenitor star has a self-similar structure with a constant envelope-to-core mass ratio  ~0.02−0.03. However, difficulties arise for very long duration (tp ≳ 104 s) shallow phases. Alternatively, a spinning-down magnetar whose emitted power refreshes the forward shock can quantitatively account for the shallow decay properties. In particular we demonstrate that this model can account for the plateau luminosity vs. end time anticorrelation.