It is not possible to 'catch immediately'. People spread it, government notices, lock-downs go into effect. A few dozen cases in a country of 1.4B is laughable.
Every single person in China who seeks medical care and who has fever or respiratory symptoms is tested.
Every person who works in a high-risk job (e.g., airport staff, border staff, some types of medical personnel) is regularly tested.
Purchases of over-the-counter fever-reducing drugs are monitored.
That's how outbreaks are caught early. If you're interested, and if you read Chinese, you can actually learn in great detail how (almost) each outbreak over the last year-plus was detected. There are detailed contact-tracing reports on most of the outbreaks.
The CN government doesn't immediately know when any one of their 1.4B citizens has COVID. Even if they did, early detection decreases death rate, it is not a cure. And zero deaths signals cure.
That's not what I'm saying. I'm saying that most cases are caught immediately.
Zero deaths really doesn't signal a cure. Low infections -> less chance of deaths. If you only have 200-300 cases a year, then there's a good chance you'd have no deaths.
So no, zero deaths doesn't mean cure, it can just as well mean very low transmission.