You mentioned that the 'main leak point inside is close to where that white pipe points down'. Whilst that video doesn't appear to show anything on the otherwise of the wall, the pipe must have been fitted for a reason. It is too high up to be an overflow or gravity fed drain from the property below, but it is angled in such a way that the person who fitted it seems to have expected water to come out. I don't know enough about the building codes for NZ to know if is some sort of approach for venting, etc? Puzzling.
When it is safe to do so could you get to the pipe and have a further investigation? An inspection camera would be ideal if you can't see the other end and have access to one. Does it look like the wall on the other side has been repaired to indicate it was once through the wall but since disconnected?
If it is worse during snow and hail then putting a physical guard (normally tensioned wires or mesh) across the entire length of roof approx. 6 inch/150mm back from the gutter edge would prevent slip down the roof getting into the gutter. A short ledge over the gutter itself that drains towards the roof would prevent it falling directly into the gutter. Those could both help in that scenario, but as it happens during really heavy rain as well it won't solve your problem completely.
If water is coming in through that gutter the cause needs addressing. You can try and seal between the gutter and the underside of the roof sheets but it won't be a permanent fix and doesn't address the cause. One temporary fix could be to clean thoroughly and gun a suitable flexible sealant into the gap between the angled top of the gutter and the underside of the roof sheet, then prime with bitumen paint and fit a strip of bitumen-based flashing product to bridge across from the underside of the roof to the side of the gutter. This might buy you some time to investigate the cause and get a permanent fix in place, but it will fail eventually.
With regards to the step, anything fitted across the internal corners to 'round it off' and reduce the friction would help. A hard flashing product could be dressed to fit such as lead or aluminium - it wouldn't need to be watertight (and likely wouldn't be) - it is just to reduce the friction and help the water to flow more efficiently.
If it rains heavily, is there any way you could get your camera/pole back up there to capture the flow? I've used a cheap web cam under a lunch box before to find a leak when access has been difficult.
When you had that short section of gutter replaced and the roof edge brought back, did the guy/persons doing it identify the actual cause, or were they just there to treat the symptom?