Shouldn't it be possible to install the mosh-server over the network from the client machine. Similar to what sshuttle does?

»Then Mosh runs the mosh-server remotely and connects to it over UDP«

i.e. yes

The documentation doesn't say this, but apparently, after you start mosh-server, it dies if there is no incoming request within 60 seconds. What is the point of the server if it is just going to quit after 60 seconds?