Does every staff member need an email address?
Yes. The email is how they sign in, so staff cannot be created without one. Any address they can actually read works, including a new free one.
Yes. A staff member's email is their sign-in identity, so there is no way to create staff without one. This is the requirement behind a stubbornly disabled Add Staff button: first name, last name, and a valid-looking email are the three things it waits for.
A phone number in the email field does not count, and neither does leaving it blank.
If you are hiring a helper who does not have an email address (a relative, a neighbor), any address they can actually read works: a free Gmail account made for the purpose is fine. It needs to be theirs, because their invitation and password resets go there, and it must not be an address already used by another staff member here.
If their email already has a NestliCare account (they are a parent here, or work at another daycare on NestliCare), that is fine too: you will be offered Send an invitation instead, and accepting from their existing login adds the staff role to it.
Related: Invite your team, Why is the Add Staff button greyed out?, Can a staff member also be a parent?.