You asked about the productivity boost... Well it's in being able to code, debug, etc... a running program.
Did you miss a decade or two of CS history? Lisp and Smalltalk have been around long enough that the value of programming with dynamic languages is known... I mean, there's shaceships and whatnot running on Lisp.
Why are you bringing up Java? Java is terrible in its own way. Of course you will be more productive without types if the alternative is only Java (which it isn't). Typescript has an extremely expressive type system.
Ah yes, don't worry about correctness at all, just wing it. I take it you haven't had to debug some of the stuff you've written?