Files missing in Solution Explorer not showing in output after build
After installing the extension, restarting VS and clean/building solution I am able to see the files that are on disk but are not included in solution in the error output - but as you can see from the screen shot missing files are not appearing in the output.
VISUAL STUDIO INSTALLATION DETAILS
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I've just done a quick test myself - modified a csproj by adding this:
<ItemGroup>
<None Include="test-file-to-delete.txt" />
</ItemGroup>
The Show Missing settings are as follows:

One thing to check - I notice you have a .gitignore file - does that mention .txt files at all?
.gitignore is not ignoring txt files - but I found this extensions as a means of tracking down some missing .html files - which it didn't find so I manually found them in the solution explorer.
My Show Missing settings are identical to the screenshot you provided - if I uncheck "Detect Non-included files" I do not get any errors in the output window.

Just noticed that the 'message importance' setting looks funny in my screenshot, and always seems to output as Error for me in 2019 - which looks like a bug, but pretty sure that's not what you're hitting. https://github.com/flcdrg/VsShowMissing/issues/105
What is the build action for your txt file?
Build action is content. My dropdown is blank too initially.
I just updated it to Copy Always and it's showing up now - is this by design? Doesn't help with static files like .js, .css, .html which are only ever moved if they're published
Hmm.. created a new ASP.NET app and added similar item to the csproj and still getting it flagged as expected.
Do you get the same problem if you create a brand new console app and add an item pointing to a non-existent file?
No, pretty sure the Copy to Output Directory shouldn't be significant.
There is some extra logging that I have for when I'm testing the extension locally (with the VS Debugger). Maybe I should add an option to optionally turn on this logging for release builds, as that might be useful for figuring out why it isn't matching your file.
If you're feeling enthusiastic, you're welcome to pull down the source, run it locally and see if you can figure out what's tripping it up :-) I wonder if there's something unique in your project that it isn't expecting.
No - the problem goes away when I create a brand new project - so there's something up with the specific solution. I'll have a go at pulling the repo and debugging locally as soon as I get a chance as I've not had anything to do with building extensions in VS and it's certainly something I'd like to learn about :-)