I think the primary reason is “population control”. It is similar to college admission that many students wanted to get into Harvard but the class size at Harvard is limited. If every self-claimed pre-med gets to submit their applications to AMCAS, med school admission offices would have to read/screen-out way too many applications, not to mention AMCAS need to verify way too many transcripts. Med school admission rates are below 10% AND those applicants already successfully pass the weed-out hurdle.
Your undergrad school name does carry some weight but it is usually in situation like 3.7 (state school) vs 3.6 (top school). However lots of students going to top school end up with horrible GPAs (like below 3.0). It is not easy to maintain 3.7 GPA in top schools (your peers are much smarter) while much more doable in average schools.