This isn't necessarily a "from a small provider" but rather "administered by the library."
It could be any email service that allows the librarian-administer to reset the password for an account. If I mess up my exchange email, the helpdesk can verify my identity and reset my password.
I suspect that most going down this approach would find it easier to use a large hosting service that they provision and administer (along with allowing password resets) than to try to have an underfunded library IT staff stand up an arbitrarily large email service and manage all parts of it.
It could be any email service that allows the librarian-administer to reset the password for an account. If I mess up my exchange email, the helpdesk can verify my identity and reset my password.
I suspect that most going down this approach would find it easier to use a large hosting service that they provision and administer (along with allowing password resets) than to try to have an underfunded library IT staff stand up an arbitrarily large email service and manage all parts of it.