Use embedded test server to get urls

This is the preferred way to get URLS within tests and might be
contributing to the flakiness. Will re-enable the test in a separate
CL since this should likely remain regardless of whether it was the
cause.

Bug: 40893068
Change-Id: I22e87ebe52f6338e2fcf31bb49569a6784408853
Reviewed-on: https://chromium-review.googlesource.com/c/chromium/src/+/6013533
Code-Coverage: findit-for-me@appspot.gserviceaccount.com <findit-for-me@appspot.gserviceaccount.com>
Reviewed-by: Caroline Rising <corising@chromium.org>
Commit-Queue: Alison Gale <agale@chromium.org>
Cr-Commit-Position: refs/heads/main@{#1382562}