Yesterday my internet service went down for over an hour but I still had cell service.
My LAN was working so I had this idea to screen cast my phone screen to my Mi box so I could maybe watch Netflix on my big screen. I went to the Google home app on my phone, found my Mi box, and selected it. After a minute or so, it failed.
Why would I need internet on my router to locally cast my phone screen to my Android TV?
You don't need it. Are you sure you were trying to cast your screen or were you casting the Netflix app?
Come think of it, in order for your phone to see your Mi Box, you would have had to be on your home WiFi, which had no connectivity. You would have been able to cast your screen, but you wouldn't have been able to load Netflix on your phone, since it was not online.
The way to get this to work is to use your phone to share its internet to your Mi Box.
You weren't connected to your home WiFi on your phone. You were connected to your LTE/4G network. Your phone needs to be on the same WiFi network as the Mi Box in order for Chromecast to work.
You're saying your phone was on the same network, but I'm assuming you were trying to stream something from the Internet. If the video was playing on the phone, you were definitely on your LTE/4G network.
If it was a local file on your phone, you may have to turn off LTE/4G in order for the phone to remain connected to the WiFi, in which case it should work.