> with far less of the "type system" enforced by the compiler and almost all of it in our heads

That's why reasonable people want to have good type systems: People who think that they can keep the type system in their head are exactly the people whose opinion should be ignored.

> Funny, but that's exactly what we Smalltalkers had in Smalltalk

Smalltalk doesn't let you say “this object responds to message Foo only when used in this part of the program”. In other words, there's no separation of concerns.

> and almost all of it in our heads

What you realistically can't produce entirely in your head is a proof that your program is correct, unless the language is explicitly designed to lift part of this proof obligation. That's exactly the role of parametricity: to help you separate concerns, allowing you to prove one small thing at a time.