Why do universities require official confirmation of scores before admission? Why do American universities want official GRE/TOEFL scores before admission? It seems to be a waste of time and money for prospective applicants. They can easily use the scores claimed by students in first stages and ask them official scores after admission. 

On the other hand ETS can send official scores directly to universities electrically via Internet without additional cost.
The problem is that many students will resort to cheating, if necessary, to secure admission to graduate school. Just about any method or loophole that can be exploited will be. For instance, when I was an undergraduate in the 90's, one of the graduate departments stopped admitting students from China for several years because they had multiple instances where applicants had stand-ins for interviews to determine their English skills. Similarly, at my current school, we have recently had a rash of applications come in with faked English competency certificates. 

Therefore, American graduate schools (and programs in other countries which ask for test scores) are increasingly (although not exclusively) relying on officially submitted documents. Scores reported directly from the College Board would presumably be less likely to be subject to manipulation. 

As far as why not just ask for a paper copy: American departments usually only have one admissions cycle per year. Therefore, admitting someone who is ineligible because of fraudulent documents usually means denying someone else who is qualified the ability to be admitted. Therefore, schools want to know that the students are officially eligible before they are admitted, and will not hold up the process until later. Schools that do rolling admissions likely do not have this problem, and could perhaps use the method you have suggested.