How important are constraints to a problem?

After all, the more solutions we have to any particular problem, the more likely we are to solve it right? In actual practice when we are properly constrained we are more likely to spend our time and resources more effectively in solving problems. An infinite number of solutions includes the bad ones, the ones that take spend our resources too freely, the ones that don't consider risks and unintended consequences, and the ones that address symptoms but do not address the root causes.

It seems like open minded thinking, but what actually happens is inspite of having a near  feasible solution, we tend to come up radical alternatives and explore them. This wastes time, resources and energy.