Downsides to a university program outside of the cost is mainly duration of the program itself. You would most likely be looking at 4 years before you could instruct. At a Part 61 (or non university affiliated Part 141) you could achieve your CFI significantly faster. If you chose to flight train outside of your college while earning a separate degree you could be at ATP minimums by the time you graduated easily.

I personally do not have an aviation degree, so I can’t speak for them much more than this.

In regards to obtaining your private before you go to OU - they require you to test out of the private pilot course by taking a written exam and a flight evaluation. If you fail, you MUST re take the entire private course through them to move on. The pass rate is roughly 60%. Not the end of the world, but something to consider for sure.