Convert all remaining uses (in api.cc) to use private local
implementations. These local uses all appear inside deprecated
functions.

Bug: v8:7786
Cq-Include-Trybots: luci.chromium.try:linux_chromium_rel_ng
Change-Id: I86bcafd4c8a737da32fcb8ab275ec708632f9e39
Reviewed-on: https://chromium-review.googlesource.com/1140319
Commit-Queue: Dan Elphick <delphick@chromium.org>
Reviewed-by: Leszek Swirski <leszeks@chromium.org>
Cr-Commit-Position: refs/heads/master@{#54514}