Fast Paced Development Model Part I

In a article I recently wrote, which was not published on this web site, I raised the question. How can a software development team of any size properly release software applications with a limited amount of acceptable bugs. By acceptable bugs I mean, bugs that do not hinder the ability of the user to properly use the application. Any takers?

The hypothetical situation I presented was:

1. Team of 1-10 developers all working on different projects at any given time but return to one main application when the need is there.
2. Application development time is 1/4 th of normal development times.
3. Development team & the companies development teams believe in documentation but never follow through.
4. No QA team. (developer is on the line to deliver on time and with a near flawless product)
5. No unit testing of any kind.

I have, to date, presented this problem to the public but none have come to a conclusion that I see will work. Would a TDD method work? Would Waterfall model wok? Or, would a Spiral model work? Or would Agile work in this case?

This will be my challenge. Find a software engineering model that works well with such a situation or propose one.