Your speculation is almost certainly correct. The charge he pled to is nonsense but he didn’t want to face the other potential charges, or have his son charged. But for those considerations his lawyers would have filed a motion to withdraw the plea which would have been granted.
The judge gave the Brady order AFTER Flynn already pled guilty, which is why we are where we are.
Flynn pled guilty Dec. 1, 2017. Judge Sullivan gave Brady instructions Dec. 12, 2017.
https://www.scribd.com/document/370672707/Flynn-Order