There are two different concepts/issues getting muddled in this discussion:
First, "traditional" telephone service vs. what AT&T gave you: In many parts of the country, telephone service providers are now using some form of VoIP, instead of classic, analog copper-wire-based connections, to provide service. In some cases, the copper wires are actually abandoned/disconnected, and in other cases, the wires are there, and could be used, but the telco offers VoIP instead. This is nothing out of the ordinary in today's telco industry. The telco is simply providing and connecting your house to their own, digital connection to their central office. In most cases, it's VoIP. In the case of fiber optic service, it may be a different type of digital service (POTS over fiber). In all cases, the utility is simply providing you with a modular jack connected to telephone service.
Using an OBi for your telephone calling is no different than, and possibly less reliable than, using a telco or cable company's VoIP service. If your neighborhood is connected to the utility via coaxial cable or fiber optic cable, then that's what the utility will use for phone service. Keep a cell phone handy if you are concerned about long-duration power outages. Otherwise, connect the in-house equipment to a UPS. Some utilities actually provide their own battery backup unit, attached near their connection to your house, and plugged into your AC power.
Second, there are two ways to have multiple telephones in multiple rooms: do as AT&T did, (if, and only if, you completely discontinue service with AT&T, AND disconnect their wiring), and connect your ATA to the house wiring. Every house phone jack can then be used to connect a standard corded or cordless telephone device. OR, abandon (do not use) the house wiring at all, and do what Leo mentioned: connect a multi-handset, cordless phone system's base station directly to the ATA. The house then uses the cordless phone handsets over DECT wireless radio.