To look at it another way, just because some code I work on at my job is derived from open source MIT-licensed code doesn't mean I personally have the right to distribute it if my company doesn't want me to. I'd guess this comes under some generic "confidential information" clause in the employment contract.

Hmm your example is different: if you manually write code, there is a copyright for it whether it is derived from an MIT-licence or not. If you don't own that copyright (because your employer does), then you don't have the right to distribute it because it is not your code.

If you generate the same code with AI, now it does not have a copyright. If it depends on an MIT library, then the MIT library has a copyright and you have to honour the licence. But the code you produced does not have a copyright (because it was generated by an AI). And therefore nobody "owns" it. My question is: can your employer prevent you from distributing something they don't own?