Nope. Everything does not work properly as you described. Yes, your devices can get online, but you cannot get to your devices. For example your Nest thermostat must connect to an external server to communicate with your phone when you are not at home. This is a bad thing, since it means that you are now relying on a third party to provide connectivity.

Also, let's say you want to connect to your Time Capsule when you are on the go. Wouldn't it be nice to be able to connect to its public static IP address directly (assuming you set your firewall to allow such connections).

Lastly, as the Internet goes IPv6 only, your IPv4 only devices will get cut off. This will happen sooner than you might imagine.

Yes, there are large swatches of unused IPv4 addresses, but the demand for new allocations is much greater. Why cling to an old system just to offset its death by a few months when we can spend the effort on making the new system work better?