All paints are toxic to a point when wet or dry. Even no voc acrylic paints are still toxic when wet or dry. The only paint that I know of that might be the least toxic is Mythic.
Oil base paints, like almost all paints, give off fumes over an extended period of time long after they have dried to the touch. This is the voc's that paints contain. Other ingredients contained in paint are toxic when the film is formed, the paint is dry.
I agree with your painter that an oil base finish will be the most durable for your trim and cabinets, easily out performing anything else. There are some good acrylic paints that come close to the same performance but with less toxicity.
Mention your concerns to the painter and ask for a friendlier alternative. All of the good manufactures have acrylic cabinet and trim paints. These dry harder than normal wall paints. Maybe one of these will work for you.