Humanity will eventually go extinct, but birth rates would need to decline much further or stay this for a very long time before it’s a meaningful issue.

Further, people alive in 2300 will be decedents of people who chose to have kids generation after generation despite living in an industrialized environment. That self selection both in terms of DNA and culture means a population bounce back becomes increasingly likely over time especially as fewer people means less pollution and less competition for resources.

Humanity might even end up cycling through industrialization, collapse, hunter gatherers, agriculture, industrialization, multiple times before settling on some stable equilibrium. You just can’t extrapolate exponential curves indefinitely when they depend on the population size.

Yes, fertility is heritable.