How to get the original search string with Google Adwords ValueTrack?

The functionality you are looking for is not available. Google documents all the available value track parameters here. From that document, {keyword} is as you describe:

For the Search Network: the keyword from your account that is matched to the search query.

I have looked through the available value track parameters and there is no available parameter for the user's entire search phrase.

The full search phrase used to be available in the referring URL, but Google has eliminated that data in 90% of cases in the last few years. Google Analytics now shows "(Not provided)" in most cases where Google used to allow webmasters to see the search phrase.