> but it's not uncommon for a coworker to check in changes to a JavaScript class that completely breaks a part of the application that the dev wasn't thinking about

You don't need to write unit tests to check the type of each value, you just need to have a decent set of integration tests to make sure you aren't breaking other parts of the app. If errors in the browser after a refresh don't help you catch it, failing integration tests will. This is how we do things on my large scale javascript project and it works out just fine; we have development challenges, for sure, but weak/dynamic typing issues is not one of them.