The flaw is inherent in the design

It seems to me that their problem has more to do with requiring a single yes-or-no vote for the entire budget than the 2/3-majority threshold. Any action taken is going to affect everyone, even those who voted against it, so it makes sense for even a small minority to be able to veto actions to which they are opposed. It wouldn't be a problem if the budget was more distributed; for example, short-term, targeted spending could be authorized alongside each bill when and if it passes. For long-term spending the yearly budget could be approved in a similar manner, and a supermajority would only required to increase it. A minority could then block specific marginal programs, but would be unable to hold the majority hostage, as it were.