But suppose A starts showing symptoms, tests positive. We can immediately notify B1, B2 and B3 that they need to quarantine, immediately. Before they start showing symptoms. And therefore we're cutting down on the time that they were infectious and spreading. Therefore generation C is only a couple of people instead of dozens. We can notify C generation as well, immediately if we wish. This causes a large quarantine - there are a lot of false positives who are quarantining without being infected - but it would massively reduce the spread. If you can immediately test all of generation B and C, you can perhaps avoid some of the quarantining (assuming a negative is reliable).
Maybe contact tracing misses some, maybe some new infected come from other countries, maybe some people don't quarantine when told. All these things keep the infection going. But it can be held down to a manageable level.