test: Port clean-distdir integration test to the modern e2e test framework (#85828)
@timneutkens pointed out that this test was sometimes flaky with the
lockfile:
https://vercel.slack.com/archives/C07UCHRBWGK/p1762267195472319
I wasn't able to reproduce this locally, but this test looked like it
was in generally rough shape, and the `integration` test directory lacks
test isolation, so this at least ports it to the modern framework. If
nothing else, the test isolation might help.