No, you can't say that. 'Staff' is a collective noun - think of 'staff' as a word which you use in the same way as 'team'. You can say 'Natalie is in/on the team' or 'Natalie is a member of the team' or 'Natalie is a team member'. You can't say 'Natalie is our team' or 'Our team Natalie', can you? That's nonsense. A team is a set of people, not an individual. Natalie is not a team - she's individual person who belongs to the team. Well, 'staff' works in exactly the same way. A company's staff is a set of people, not an individual. Natalie is not a staff - she's on the staff, so she's a staff member or a member of staff.

Su.Ki’s strong answer is correct, but DavidK is right, too — we can refer to Natalie B. as “staff” when staff is treated as an adjective. The contexts I mostly think of where we’d use that are dismissive — “Natalie is just staff.” But — “Natalie B. isn’t available then. She’s staff, and they have the night off.” But the general statement “The staff have the night off [and that inherently includes Natalie]” is more natural.

“Staff” is used in a way not mentioned here, possibly regionally. Natalie is on the staff. (Standard) Natalie is staff. (Variant with the same meaning. To distinguish her from belonging to another group, such as “client”, “faculty” etc. But you can’t go further and say “staff Natalie”. You need to say “staff member Natalie”)

I don’t think so, “Staff” is like “team”. :)

It may be done, but generally, no. A member of our staff, Natalie, .... One of our staff, Natalie, .... Natalie, on our staff, .....