A budget that can support the agency fully realizing their concepts in-house. For me the main motivator of producing something quickly and “seeing what they think” is that the budgets I’ve worked with are just too small to warrant any experimentation. There just isn’t room to follow rabbit trails aren’t a sure thing.
Utter respect and trust that the agency will represent you in the best possible way. The respect that Avis has for DBB is clearly illustrated in #1 and #4:

“Avis will never know as much about advertising as DBB, and DBB will never know as much about the rent a car business as Avis.”

“To this end, Avis will approve or disapprove, not try to improve, ads which are submitted. Any changes suggested by Avis must be grounded on a material operating defect (a wrong uniform for example).”

Most clients I’ve met with just wouldn’t be comfortable with that much of a hands-off approach.

But in the end, I do think that you may have compared apples to oranges here with the 37signals quote. No matter what the budget size, 37signals strategy is the best for product development. When you are launching a long-term software project, there is no rush to perfect it. You actually get more good press and PR by updating often. But when you are doing creative services and marketing, there is no room for launching something that’s not ready for prime time!