Scope, Schedule and Quality define "What" the project should deliver, however they don't define "How" it will deliver. As stated in other answers, more budget can allow the "How" to be flexed. In my mind, I see the "Golden Triangle" as more like a "Golden Pyramid", where the base triangle is formed of the "Cost, Time, Scope" constraints, and the height of the pyramid defines the "Quality". Hence all four are interlinked.
It is true that throwing resources indiscriminately at a project is seldom effective, however if the timing is right, more resources may be able to bring in a project on time and to quality. How? - well, there is no single right answer, but breaking the project into more manageable chunks and allocating the right resources to the right elements of the project will generally be better than just chucking more developers into the mix without restructuring the project to accommodate them.
Alternatively, more funds may also allow a different approach entirely. For example, rather than building a solution in-house to avoid the cost of buying an expensive yet commercially-available solution, the extra money may change that decision. You will still provide the expected outcome - but you will buy the solution rather than building it. But once again, if you wait until too late in the project, you'll fail.
What I am saying, then, is that if you identify issues early enough, you can take steps to spend more money effectively to deliver the project. And that's the key: timing is everything. Crossing your fingers and hoping it will "be alright on the night" until it becomes clear that it won't, is a recipe for disaster. Proper project management, tracking, monitoring, risk analysis, planning, and conversations with your project sponsor will always pay dividends, and allow you to make changes early enough to be effective, by identifying issues at the earliest possible point, then spending the necessary money (or changing other aspects of your approach) to address the emerging issues.