@Much2learn, states may not want to fund their publics, but that doesn’t mean that they want to give up control of their publics.
Yes, many publics have constraints if you want to change in to their most popular majors. One way to look at it is that they are maximizing their resources (and the elite privates have the luxury of maintaining excess capacity). UIUC is also a bit on the extreme end in fitting kids in to majors (even among publics), for better or worse.
Have you looked in to Engineering Pathways?