Having worked in hotels (not Disney), you have a better chance of getting the room type you want the earlier you check in. When you check-in early ahead of most other people checking in that day, they still have most of the available room inventory to work with and can switch some thing around to accommodate your request.
The rooms given away earliest in the day aren't the "least desirable", they are the ones housekeeping cleaned first in their schedule, and I know at Disney they pre-assign rooms so they are assigned ahead of time according to arrival times of online check-in.
"Upgrades" are more likely to happen later in the day (if at all) and typically happen if they oversell on room types and all the rooms have been distributed. They will "upgrade" you if they run out of those rooms. Similarly, if the hotel is oversold entirely, you run the risk of being walked to another resort the later you check-in. This is a rare occurrence however.
My opinion, the earlier you check-in, the more options you will have.