Having an LLM helps, especially when you're facing a new subsystem you're not familiar with, and trying to understand how things are done there. They still can't do the heavy duty driver work by themselves - but are good enough for basic guidance and boilerplate.

If you use AI to make repetitive tasks less repetitive, and clean up any LLM-ness afterwards, would they notice or care?

I find blanket bans inhibitive, and reeks of fear of change, rather than a real substantive stance.

As long as they align with the correct (i.e. yours) values, of course. When they adopt the wrong values, it's not fine.