How can I get the contents of the VS 2022 build output pane through the given DTE2 object in Powershell?

Question

I have a little program I wrote using the technique mentioned in a lot of places on the web to obtain the DTE2 object for a running VS instance. It is available on github - https://github.com/MarkKharitonov/RunningVSInstanceFinder. I can use it to obtain the DTE2 object from Powershell:

C:\work> git clone -q https://github.com/MarkKharitonov/RunningVSInstanceFinder
C:\work> cd .\RunningVSInstanceFinder\
C:\work\RunningVSInstanceFinder [master ≡]> dotnet build -v:q -nologo

Build succeeded.
    0 Warning(s)
    0 Error(s)

Time Elapsed 00:00:01.06
C:\work\RunningVSInstanceFinder [master ≡]> $VSInstanceFinder = "C:\work\RunningVSInstanceFinder\RunningVSInstanceFinder\bin\Debug\net472\RunningVSInstanceFinder.dll"
C:\work\RunningVSInstanceFinder [master ≡]> add-type -Path $VSInstanceFinder > $null
C:\work\RunningVSInstanceFinder [master ≡]> $res = [Ceridian.RunningVSInstanceFinder]::Find("C:\dayforce\CSTool\CSTool.sln")
C:\work\RunningVSInstanceFinder [master ≡]> $res.DTE.Solution.FullName
C:\dayforce\CSTool\CSTool.sln
C:\work\RunningVSInstanceFinder [master ≡]>

Now I want to grab the contents of the build output pane, which is active:

But nothing I tried worked:

1. Some sites suggest to inspect the ToolWindows property, but it is empty:

C:\> $res.DTE.ToolWindows
C:\>

2. Some (e.g. How to get Visual Studio Output Window content via EnvDTE (Non Package)) suggest to leverage the Windows property to find the right window, but it does not work either:

C:\> $o = $res.DTE.Windows.Item([EnvDTE.Constants]::vsWindowKindOutput)
C:\> $o
System.__ComObject
C:\> $o.Object
C:\> $o.OutputWindowPanes
C:\>

I know the DTE2 object I have is good, because I can trigger a build in the VS instance:

C:\work\RunningVSInstanceFinder [master ≡]> $res.DTE.Solution.SolutionBuild.Build()
C:\work\RunningVSInstanceFinder [master ≡]>

And it indeed runs the build and the build output pane reflects that.

At this point I think to move the investigation into the C# code, but I would really like to understand what is going on in Powershell.

How can I get the Build Output pane content from VS 2022 in Powershell?