Totally agree with you that if you put headlights on and then crank it might be possible to see how much charge there is in the battery. But the majority of people broken down tell us the battery is ok because the lights work. They don't realise the massive difference in current required to turn an engine as opposed to running lights.
What we get normally these days is that people suspect the starter motor when it is just a discharched battery. The starter chatters because there is enough power to engage the starter motor but then the voltage drops so the starter releases and is does this rapidly and you get the rapid clicking.
A heavily sulphated (or deeply discharged) battery will normally recover but may need a bench charge.
The Jazz doesn't suffer with connection problems or starter related problems.
The other problem we get with all petrol cars is flooding, this happens if the engine is started when cold and only run for a short period, maybe and minute or two, then turned off and left until much later or the next day.
The Jazz is one of the best for reliability. Any breakdown is always down to human error, although some of the early Jazz batteries and exhausts are starting to fail.