[I'm posting this as a new answer rather than editing the original because it's completely different. Replacing or appending to the original answer would make the ensuing discussion confusing.]
What you need to do is use a regex to extract the values from the Readings property of the output of Get-Counter, and manually construct CSV output from the timestamp and those values. Change the last line to this (format according to your preferred style):
| %{'"' + (Get-Date $_.Timestamp -f 's') + '","' + (([regex]::matches($_.Readings, '(?<=\\\\.+?:\n)(.+?)(?=\n)') | select -ExpandProperty Value) -join '","') + '"'} | Out-File $file -Append -Encoding ASCII
To break that down:
(Get-Date $_.Timestamp -f 's')
This part is not strictly necessary, though I think it will make your results easier to follow. The 's' format puts the date in an ISO 8601 sortable pattern. You could substitute 'u' for another sortable format, or use your favorite custom format string. Or just replace it with $_.Timestamp to retain the original format.[regex]::matches($_.Readings, '(?<=\\\\.+?:\n)(.+?)(?=\n)')
The regex matches the contents of any line that is preceded by a line that begins with \\ and ends with : (those pesky counter names you wanted to get rid of). Note that I'm using [regex]::matches, which performs a global match, as opposed to [regex]::match or -match, which will just give you the first match for each string (the Readings property is a single string, so only the first counter reading would be returned).| select -ExpandProperty Value
Produces an array of all the matches, which you can then join with "," and surround with "'s to produce CSV output.
Since you're not using a conversion function, you also need to construct a header row. Add this line right above the pipeline:
`'"Timestamp","' + ($p -join '","') + '"' | Out-File $file -Append -Encoding ASCII`
That's assuming that $p is an array (which it should be). If it's a string, then depending on the format you can either use it as-is, or -split it and rejoin it in CSV format.