I have an update: We took the console to a local repair shop with a good reputation, and they determined that the HDD was bad. I told them that was impossible as it was brand new. We picked up the console and did a full restore. We did not do a system transfer, instead we signed into it as a new console. We used the Playstation Plus service and downloaded the save data from the cloud and reinstalled the games. As of me writing this, it is playing just fine with zero crashes! My theory is that a corrupted file must have just been transferred over, and that is what was causing the crashes.
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