"stable" here means zero population growth over the long term, which actually requires a little above 2 per couple to account for people who die before being old enough to parent children. 2.1 is the common approximation for that.
But people live after they reproduce. Having 2 children, and every 2 in the next generation have 2 more, etc, means population continues to grow. A steady state is reached, but as lifespan increases so does population. The obvious limit is, if lifespan is infinite, then 0 children is the only number that permits a stable population.
The average human lifespan is a relatively fixed sized buffer. Since everyone who is born eventually dies, input matches output, and there's no population increase other than the average lifespan (and thus the buffer size) increasing.
