I wonder, how can a programming language have the productivity of a high-level language ("write like a high-level language"), if it has manual memory management? This just doesn't add up in my view.

I'm writing my own programming language that tries "Write like a high-level language, run like C.", but it does not have manual memory management. It has reference counting with lightweight borrowing for performance sensitive parts: https://github.com/thomasmueller/bau-lang

C is literally a high level language.