You would think that it will save you a lot of money if the school ask for official school only if they have intention to admit you, and it shouldn't be difficult to do for the grad schools.
Well, indeed this can be done, and is being done by a lot of programs. However there are complications and many programs don't do it that way.
First of all, you get four scores to be sent out for free when you take the test. Even if you apply to as many as 10 schools, that may be enough to cover all the schools that do ask for official GRE/TOEFL scores upfront. This is intended to be a primary channel through which you are reporting your scores. It will be a significant cost if you apply for 30 schools, but I don't really see the point of doing that. Of course, that would require you to do your homework early and figure out which school to send your scores to.
Second, that adds one more phase of admission process. You have to understand that by easing your financial burden (which isn't even the main cost of application anyways. just think about how much application fees and tests would cost you.), schools have to put in more work, with no apparent gain on their part.
If your financial constraints are preventing you sending the scores, you can always write a letter to the program you are applying for. As an international graduate student, I can tell you that many institutions did give me waiver for official scores before admission, and some even waived application fees entirely.
At last, in some universities, there are two separate entities that are processing these applications: the graduate school and the department. Sometimes the graduate school have minimum requirement policies for GRE or TOEFL scores and until you can prove your eligibility, your application cannot even be transferred to the department, where the decision is actually made. If that is the case, it would be impossible to waive unofficial score until admission.