> It makes more sense when you read it as "Windows subsystem for [running] Linux [applications]".

You can't have ellipsis when the shortened version already has its own meaning.

X for Y when both X and Y are nouns means that X is part of Y, not that Y is part of X.

e.g. "I bought new tyres for my car". The tyres are part of my car. You can't flip it and say "I bought new my car for tyres", it's just not how the word "for" works.

Grammatically it has to be "Linux for Windows subsystem", or "Windows subsystem for running Linux" as you said. The verb is essential for it to parse correctly.