Heck, all you need for proof of that working is to scroll up and look at the people giving Zappos free advertisement on HN.

Yeah, but the return on investment in marketing is likely much, much higher. Or, better yet, invest in making your product better so customers don't need to call support in the first place.

Customer churn is also very, very hard to quantify, and your customers will leave you for a variety of reasons, many of which would not be impacted by better support (i.e. they move, go out of business, get purchased, etc.)

Yes certainly, it is an illusion that marketing, customer support, product development and sales are all discrete things without any overlap.