Just bumping to say, this is what fixed it for me. I have two Win11 machines, one where lightshot was broken and the other (newer!) where it worked fine. Again, the app was not showing in the system tray, so there was no way to access settings or use an alternative method to trigger screen capture, there was simply no way to interface with the process at all.
The difference was the Intel Arc graphics driver, lightshot was back where it belongs upon reboot after installation. Hope this helps someone else.
Edit: my machine in question is a Razer Blade, with nvidia's Optimus. aka, it switches dynamically between integrated and discrete graphics processors/display adapters based on load/demand. I didn't have to fiddle in any control panel, the Intel driver & support assistant simply told me to update the Intel graphics driver, and that solved it. As an aside, that driver (and the two wireless drivers it asked me to update) do NOT show up in the "optional updates" section of Windows Update, no matter how many times I refresh it and check.