Automatic moving websites due to server load should be optional, in my opinion.
The reason: I put a website on a server for a particular reason. For example, I have SSD servers and NVME servers, and some are storage servers using HDD. Also, some specific servers are for fast PHP CPUs with high-speed, matching RAM.
Others are average, according to price.
So, moving from any server to another server only because the lower load is not suitable in most scenarios, except if you host all clients with the same package on a server group.
The idea is very good but must be implemented with proper options and needs to consider scenarios.