[chore] remove env GH action and browser caching

[Conduitry]

.github/actions/env has been bugging me for a while. We had built code that needed to be checked in - and which apparently hadn't been kept up to date in a while, because whenever I ran pnpm build locally, it changed this file.

We'll see what happens when CI runs, but this attempts to replace that whole file with a re-creation of just the functionality we need, which lets us drop the build process for the file and remove the whole @actions/core dependency.

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[Conduitry]

After #4727 is resolved, I'll merge master back in here and we'll see what happens.

[Conduitry]

This now removes the env action entirely. pnpm caching was getting handled another way now, and the Playwright browser caching was of limited value.