I can add an IP as a loopback just fine - my routers have that all the time.

The problem is then how does the network know where my IP address is. When a packet aims at my loopback, how does it work its way to me?

The answer is the same as interface addresses - we use routing protocols for that, mainly today that's OSPF (in local networks) and BGP (globally).

Convergence takes a relatively long time for BGP, and BGP (at least at a global scale) is limited to /24s. ISPs won't accept your /32 advert.

Even if they did, traffic still only goes in one direction from a given host. On top of that, theoretically if every BGP peer rand BFD you might see failovers in a few seconds, but that's not the real world.