Thoughtful question . . .
Ann Arbor has no moratorium on approval of newly-registered rental housing units.
Regulations are uniform city-wide, and are focused on issues impacting the safety and general habitability of the rental units. There is a requirement that newly-registered units be inspected; those that pass inspection are granted a three-year certificate of occupancy.
Ann Arbor has a very active, competent department devoted to providing services to both tenants and landlords. It is called "Rental Housing Services", and here is their webpage:
Contact person in the department is: Kelli Johnson, Administrative Specialist; (734) 794-6000, x42667.
Hope this information is helpful to you.