As far as I am aware, the only speedlights which can operate without any internal batteries are the Metz range. All the others need those 4 pesky AAs inside the flash at all times for control. The external HV power pack bypasses ONLY the internal supply to the capacitor.
Hence, if you change the speedlight to "External Power only" you are merely cutting off that portion of the recharging power drawn from the internal AAs. I assume that there is no indication on the power pack of recycling (change in status lights in one form or another is a popular way of showing this - some of my Pixel TD-381s twinkle while recycling, the newer ones flash on/off more slowly then return to constant when fully recycled. My CL4500-equivalent, the Godox PB960, ramps up from no blue lights to (eg) 3 blue LEDs then the red ready light comes on)
One way you could check is to set the speedlight to "External Power ONLY" and manually fire it with the test button. Fairly obviously, with only the HV pack as power source if the flash fires and recycles then power IS getting through.
One potential area to check for faults is the cable. Though these things look sturdy on the outside, internally each of the 3 individual wires usually comprises 7 ultra-thin strands of copper thinner than baby hair. Easily broken. VERY easily broken. Can you test the 3 individual wires for continuity with a multi-meter? Test both with the cable relaxed and with it stretched to normal working length. It's very common indeed for cables to be over-stretched and then you get intermittent breaks (usual culprit is pulling the plug out by the cable instead of the plug itself) - trouble is that these intermittent breaks don't always show up on test unless you replicate the conditions of use, which includes stretching them.