Remove use of global clocks in TestClient

The functions that are relevant here can be abstracted away with
WaitUntil, which is available since this is a test-only target. This
may be updated once the ClockVariant is simplified to relevant clocks
for testing.

Bug: webrtc:42223992
Change-Id: I68a0514be56931e44777f4cd086f5ea794da377a
Reviewed-on: https://webrtc-review.googlesource.com/c/src/+/407405
Reviewed-by: Tomas Gunnarsson <tommi@webrtc.org>
Commit-Queue: Evan Shrubsole <eshr@webrtc.org>
Auto-Submit: Evan Shrubsole <eshr@webrtc.org>
Cr-Commit-Position: refs/heads/main@{#45529}