The first step is to confirm that you have the proper setup to use Universal Control. You have two Mac mini desktops, each must be connected to its own display. Universal Control is not intended replace or function as a keyboard/video/mouse (KVM) switch, but that is not the case. Rather, Universal Control is only about 'control' – the keyboard and mouse part of a KVM, but not the video part. Each device must have its own screen.
Here is Apple's description: "When using Universal Control, each device shows its own screen and apps, but you can use a single keyboard, mouse, or trackpad to move the pointer, type, and even copy content between them."
From: Universal Control: Use a single keyboard and mouse between Mac and iPad - Apple Support
Assuming that each Mac mini has its own display, then some troubleshooting is in order.
First, note that in addition the requirements you listed as properly satisfied above, neither Mac can be sharing its internet connection (both must be directly connected to the same Wi-Fi network).
Steps to restore Universal Control:
When looking at System Settings > Displays before Universal Control is active, you should see options to Link keyboard and mouse to and also to mirror or extend display to compatible devices
Universal Control is working properly for me on macOS Golden Gate.
For example, here is what I see on my personal MacBook Pro, which is in clamshell mode with my work MacBook Pro (name redacted) open on the left side of my desk and my iPad Pro open on the right side of my desk.

Macs/iPads show up for both universal control and screen mirroring, AppleTVs show up just for mirroring.
Note that in Advanced, I only have two of the options selected, not the one to automatically reconnect (which is why the other two devices are not already connected).

After connecting both devices by 'pushing through the left and right sides' of the LG display, both show as being under Universal Control, and I can move the mouse across all three displays on my desk.
The Displays settings pane shows all three devices (they can be rearranged in the Arrange settings), and a Universal Control menu bar item appears that allows me to easily disconnect one or both.

