Would we still use private LAN addresses?
Just to clarify, there are "private" addresses. There are unique local addresses, which are similar in function to the RFC 1918 addresses in IPv4. However, they are not normally used for NAT. There are also link local addresses, which have no equivalent in IPv4. Every IPv6 capable device has one, usually based on the MAC address. It's what's used for router addresses. It's also used for other things that are confined to the local network. For example, Windows HomeGroup networking uses it. In fact, it won't even work over IPv4.