Right now all the schools that are 'University of' are under 'U'. For example, 'University of Florida' is under 'U' instead of 'F'. Should 'University of Florida' be under 'U'?
In short: No, because it makes it harder for the user to find an entry in the list.
In long: Many alphabetised lists use a slightly non-standard collation, in order to make the lists easier to use.
Example 1: When listing people, they are almost always sorted by surname (Smith, John, Mr) even though the surname is the last name as normally printed (Mr John Smith). This is because in the context of finding a person in a list, the surname is the most important bit of data.
Example 2: When listing surnames (e.g. in a directory), any starting with Mac or Mc are grouped together; this means that a user who isn't sure whether they're looking for MacIntyre or McIntyre only has to look in one place to find them.
Example 3: When listing musical band names (e.g. in a music shop), there are so many starting with "The" that the T section would be unhelpfully large. So for example The Beatles are found under B, and The Who are found under W (yes, really).
Example 4: When listing lochs in Scotland, there are so many beginning with "Loch" that the L section would be unhelpfully large. So for example "Loch of Aboyne" and "Loch Achray" are found under A.
In the case of universities, it's unhelpful to group them all under U because:
A real life example from UCAS has the following list:
I'd say YES to your title question. They should be grouped under U. The institutions would rely on their name as a brand and it wouldn't make sense to break up a brand.
More analytically thinking... The separation that you're asking about is similar to the word "The" and perhaps others that prefix so many things.
For argument's sake, would it make sense to re-list all of those universities as, for example:
The last point is a perfect case where it may break down.
Should "University of the Pacific" be listed under P, T, or U?
The answer should be U.
If the band "The Who" had an public office, and were listed in the phone book, they'd not be "Who, The". :)
No. If you go to a music stores, bands like 'The Who' and 'The Doors' are never slotted under the letter 'T'. I suggest that you disreagard prefixes like "School of...", "University of...", "Institute of..." etc.