In my experience, SideCar provides a very poor experience. (I’ve never tried their product, but at the moment I’m writing on an iPad Pro and I’m VNCing into my Mac using Edovia Inc’s Screens, which is an excellent iOS/iPadOS VNC client).
(That’s ignoring the new weekly crashes that the virus/firewall scanner now causes... which is another story).
Your use case seems to be different.
I've been trying the built-in screen sharing VNC, and that too is very laggy.
The Mac mini has tested fast (iperf3.. latency and bandwidth are good) over WiFi, so I think an app has the potential to be fast.