"Overwintering chillies vs discarding them evoked some strong views on an episode of Gardeners Question"
Oh, I must have missed that episode....what was the 'strong views' about it?
"once the plants have fruited, do you keep them on for further seasons, or do you start again from seed each year?"
Bit of bot in my 'case'. What's worth it to save ,I save and what's nothing special I'll let them pass. There is such a amount of varieties out there to try that I don't waste effort for those that are not worth it. I might save some seeds even from those that I don't over winter.
Some chillies that I over wintered has fruit on bushes right now, not ripe for while yet, but plenty on. Although I've sown a lot of chillies back in January and had some over winter as well, I've just sown some more, purposely late for over wintering. They are not going to manage to fruit and I'm not expecting them neither, but hopefully make good strong plants. The bonus with over wintering is that if you manage to save them..following years the bushes just get better, bigger and crops huge..not just odd fruit like with some of the first year chillies but fruit by almost bucketfull quantity. I've got napia plant from last year that is now 3ft high and wide and it is still growing before it starts fruiting..it certainly has 'frame work' to carry plenty of fruit. Lot of people don't realise that chillies can be kept going for yeeears and made into bonsai form as well if wanted = bonchi.
As for your "mexican restaurant" chilli seeds...they could be anything..there is just a variety of possibilities that it is impossible to guess. We can identify plants for their species by the green growth and flowers, but even with ripe fruit it can be difficult pin point to exact variety. Unless it is something very distinctive. Even the jalapeno is not just jalapeno..there is many varieties that look identical but the difference between varieties is only something minute like how many days it takes to plants to mature to crop.