As others have said, the stop/start issue also occurred on my 69 plate Jazz which I picked up in 2021 (straight from the dealer as a 12 month old ex-demonstrator with the original OEM battery and 3,500 miles on the clock - so not messed around with to the best of my knowledge). I did lots of web research and have read more threads than I would care to mention. At first I found the whole thing frustrating and a bit worrying, but I soon discovered that this is a common issue for the Honda Jazz.
The stop/start seems to work for me only during Spring and Summer - I can never recall it working from late Autumn onwards when it gets colder. At the two last Honda Services (at two different Honda Dealers and both in December), I asked the service department advisor about this and on both occasions they said that it doesn't work when the battery is below a certain level of charge (my battery voltage was recorded as 12.3 and 12.6 on each occasion, which they said was normal).
However, please bear in mind that I don't possess the mechanical or electrical knowledge that other members have so you may wish to take my experience with a pinch of salt.
Personally, I no longer worry about the issue - the car drives nicely and still returns a better mpg than my previous Mini Cooper Diesel. I also take some solace from Scotty Kilmer saying on his YouTube channel that the start-stop function isn't necessarily a good thing anyway in terms of additional wear & tear on the engine from constant stopping and starting.
