Both the flash tube AND also the charge capacitor can go bad, the light duty charge capacitors found in compact and in-camera flashes are highly susceptible to expiring from overheating when run at a high duty cycle like you described. They generate significant heat when charging and discharging and can't stand up to a high duty cycle unlike studio flash units which have physically larger capacitors designed for this type of service and most also have forced air cooling.
A capacitor which has deteriorated due to heat will no longer be able to charge to the desired voltage causing the charge circuitry to remain in a continuous charging mode with the flash not functioning.
If you try to tackle this yourself, use caution both for the camera and yourself. Even if the capacitor bank is the issue, a failing capacitor can still store a significant charge because the voltage from the camera battery is raised to several hundred volts to "fire" the Xenon flash tube when triggered via its excitation input. It isn't what I would consider a do it yourself project unless you already have experience working around cramped circuitry that generates and uses high voltage. If you are considering doing this, leave the camera with the battery removed for several days and then SAFELY discharge any storage caps in the flash area via a resistor rather than just shorting the capacitor which can cause damage to it and other components.
There are many ways to hurt yourself and/or the camera and unless you have knowledge and experience in this area then it is best left to a proper service center. I have repaired a lot of complex electronics items and I am comfortable around high voltage but that is because I understand the circuitry AND I take proper precautions, neither is optional. A charged capacitor in a strobe circuit will expose you to sufficient voltage to overcome skin resistance with the ability to easily supply sufficient current to stop the heart muscle or even worse cause arrhythmia. I would use the same caution working on one of my Canon speedlights or Hensel studio strobes as I do with high power tube type transmitting gear because either can be easily lethal.
Rodger
EOS 1DX M3, 1DX M2, 1DX, 5DS R, M6 Mark II, 1D M2, EOS 650 (film), many lenses, XF400 video
