Create a dummy display for the full-screen presentation, share that into the remote meeting/projector, and preserve the primary display for presenter notes, logtails, chat, etc.
Bonus to sharing only a window - you never accidentally show something you didn't want to, like that email or messages notification...
I've given up the idea of connecting to mini over screensharing and connected a couple of displays to it, happily.
If you connect a Mac (mini, MacBook pro, air) to an external monitor, AND this monitor is not recognized as "retina" (eg. Apple $4599 XDR) AND you are not running at the monitor's highest resolution, the screen will be blurry.
There are a lot of "ands" in the above sentence, but it's actually quite common situation. Retina monitors are expensive. For non-retina monitors, native highest resolution still renders objects too small for many people.
On MacBook Pro/Air the solution is to mirror the screens. The MacOS thinks that it renders to the retina screen and sends appropriate re-scaling to the external monitor. But mirroring has its problems. For example, expect ratio will be the same as MacBook's internal screen. This may result in black bars on the monitor, depending on its aspect ratio.
Also, in M1 MacBook Pro 2021, the aspect ratio is variable, due to the 72 pixels menu bar up top, going in and out. I am not sure what will happen when you mirror the screens there.
Hopefully the above software solves this problem.
Mac mini's without a monitor hooked up can only display at a certain aspect ratio while remote screen sharing. The typical solution is to use a dummy hdmi plug (~$10) plus an app like SwitchResX (~$20) to support custom screen resolutions.
If OP's app works as described, then this is a free software solution to a $30 problem.