Why do we need live migration support in hypervisors or cloud providers?

If a application is developed as a REST API based with separate UI and backend tiers, and we have tools to deploy the servers in a few minutes, why do we need live migration support?

Say to move from one region to other, we can simply create a new load balancer in the new region and then create new VMs over there and just switch the DNS address to new IP. It's done right? So why do many companies look for live migrations?

Not all applications support load balancing as you've described, and not all companies are large enough that it's worth running that number of servers to get that type of redundancy. If the company is small enough, they may only have a single server.

Live migrations mean essentially zero downtime for these types of applications, no manually building up additional copies of infrastructure/VMs (many companies don't have the level of automation you strive for), and no shutting down existing servers.