I think choosing camera equipment is an artistic decision that should not just be picking what’s state of the art at the time.
The equipment that a movie is shot on is going to have an impact on the final product, no matter what. Most artists get this, but they don’t take proper advantage of it.
I think recording a movie with vintage equipment, or mimicking their output, can have a psychological impact on the audience that elevates it to the level of great films of the past.
Does anyone know any examples of this being used?
Movies are still shot on film if that’s what you mean. What is and isn’t state of the art is subjective.
No, more like the differences in lenses, how they pull focus, the light they let in and f stop for the shutter.
The film Matters too, but since OP specifically asked about vintage equipment, film was a given