It may , but if I were to ordering a meal in a café and I could see two people waiting to cook and one was quite obviously dirty or similar, I may say (rudely but honestly) 'I'd rather not have her cook(my meal)' whilst gesturing to who I meant.
One of two things when you hear this said:-
1. I'd rather not have her cook- because she's a terrible cook and her food is awful.
2. I'd rather not have her cook, because she does so much already it would be nice for her to have a meal she does not have to make herself.
third option....an employer is talking about taking over someone else’s kitchen staff......says "well I will take the waiters but I’d rather not have her cook".
It's an Americanism. In UK English we might say "I'd rather she didn't cook,", or "I'd rather not have her cooking."
Not wrong, just not ordinarily used in Britain. But the way things are going with the 'standardisation' of our speech with American, it soon will be...