Migrate run_testserver.cc to always use the EmbeddedTestServer.

It was using the SpawnedTestServer, which is deprecated, for WebSockets.

Bug: 380704159
Change-Id: Ic37db3000a1951a30a116c27ca9df5bef28c3bab
Reviewed-on: https://chromium-review.googlesource.com/c/chromium/src/+/6710725
Commit-Queue: mmenke <mmenke@chromium.org>
Reviewed-by: Adam Rice <ricea@chromium.org>
Cr-Commit-Position: refs/heads/main@{#1484910}
NOKEYCHECK=True
GitOrigin-RevId: 63b340cbcbf84cb5e64c44a2da5178eb575b80e4