Is there a way for me to cast my Android phone to Solus Budgie like I cast it to my Android TV?

scrcopy might be of limited use to you. You can mirror your phone's screen to your computer. The phone screen needs to remain on interact with it.

It's available in the Solus repo.

More here

@devmrfitz, I may be wrong about having to leave the screen on. It's a long while since I used scrcpy. I noticed this on the scrcpy site . I haven't looked further into it.

Turn screen off
It is possible to turn the device screen off while mirroring on start with a command-line option:
scrcpy --turn-screen-off
scrcpy -S
Or by pressing Ctrl+o at any time.
To turn it back on, press POWER (or Ctrl+p).