PowerShell Tee-Object not capturing debug lines in file

Unfortunately, PowerShell's tracing output (activated via Set-PSDebug) operates outside PowerShell's system of output streams, so such output cannot be captured from inside a PowerShell session.^[1]

However, an outside caller of PowerShell's CLI^[2], such as cmd.exe in your case, does receive tracing output via stdout (the standard output stream)^[3], so you can use cmd.exe's redirection to capture everything in a file and print it to the console afterwards.

A simplified example (from cmd.exe):

C:\>powershell -c "Set-PSDebug -Trace 1; Write-Host 'Hello, World!'" > output.txt & type output.txtDEBUG:    1+ Set-PSDebug -Trace 1;  >>>> Write-Host 'Hello, World!'Hello, World!

^{[1] Perhaps surprisingly, despite the word "Debug" in the cmdlet name, tracing output does not go through stream number 5, the debugging stream, the way that Write-Debug output does.}

^{[2] Note that you can use this technique of calling the CLI, invariably as a child process, from inside an existing PowerShell session as well, in which case that existing session acts as the outside caller. However, note that passing the command to execute via a script block ({ ... }), which is normally preferable, would not work in this case, because a script block-based invocation uses PowerShell's remoting infrastructure behind the scenes, which preserves the PowerShell-specific output streams (and also type fidelity, with limitations), so that the tracing output again goes straight to the console.}

^{[3] Perhaps surprisingly, all of PowerShell's output streams - as well as to-console-only output such as produced by tracing - are by default mapped to an outside caller's stdout stream - including errors. While you can at least separate out errors on demand via a stderr redirection - 2> - on the caller side, all remaining streams are invariably sent to stdout - see the bottom section of this answer and GitHub issue #7989 for more information.}