Observations suggest that mature faults host large earthquakes at much lower levels of stress than their expected static strength1-11. Potential explanations are that the faults are quasi-statically strong but experience considerable weakening during earthquakes, or that the faults are persistently weak, for example, because of fluid overpressure. Here we use numerical modelling to examine these competing theories for simulated earthquake ruptures that satisfy the well known observations of 1-10 megapascal stress drops and limited heat production. In that regime, quasi-statically strong but dynamically weak faults mainly host relatively sharp, self-healing pulse-like ruptures, with only a small portion of the fault slipping at a given time, whereas persistently weak faults host milder ruptures with more spread-out slip, which are called crack-like ruptures. We find that the sharper self-healing pulses, which exhibit larger dynamic stress changes compared to their static stress changes, result in much larger radiated energy than that inferred teleseismically for megathrust events12. By contrast, milder crack-like ruptures on persistently weak faults, which produce comparable static and dynamic stress changes, are consistent with the seismological observations. The larger radiated energy of self-healing pulses is similar to the limited regional inferences available for crustal strike-slip faults. Our findings suggest that either large earthquakes rarely propagate as self-healing pulses, with potential differences between tectonic settings, or their radiated energy is substantially underestimated, raising questions about earthquake physics and the expected shaking from large earthquakes.