Very seldom is a broken boiler past repair. Boilers periodically breakdown, as do other common home appliances.
If a tumble dryer breaks down you might consider repairing it. However, more often than not, it is not economically viable and due to the fact that it only needs to be plugged in, a new one is often the best option.
In the case of boilers they fall into the category that we call fixtures. There is a lot of work to replace a boiler, a lot of labour, a Powerflush and last but not least the cost of a new boiler itself. It is not uncommon to hear a quote for a new boiler to be in the £4000 range.
I like repairing boilers. I have a lot of experience with all makes and models. So long as spare parts are available why not repair it. There does occasionally come a time when a boiler is past economic repair, but not often. Of course new boilers are more efficient, but for a lot of people they need to budget over a period of time to replace a boiler, whereas the cost of a repair is much more affordable. It could be a small part worth £20 and a couple of hours labour, it could be a major part such as a heat exchanger but still only a fraction of the cost of a new boiler.
Servicing, is also a major part of keeping a boiler alive, efficient and safe. The recommendation is to have an annual service, this is when faults can sometimes be identified before they cause a breakdown and more importantly, dangerous developments show up. Once I have serviced a boiler I add it to my list and send out an annual reminder.