Revert "Use internal Picker search field text on chrome branded builds."

This reverts commit 4aac1b8191efc99f122a7529d6a7594e893e2430.

Reason for revert: Causes test failures on linux-chromeos-chrome,
e.g. https://ci.chromium.org/ui/p/chrome/builders/ci/linux-chromeos-chrome/40403/test-results?sortby=&groupby=

Original change's description:
> Use internal Picker search field text on chrome branded builds.
>
> See also https://crrev.com/i/7103935 which adds the string definition.
>
> Bug: b:327983570
> Change-Id: I3d555f89a3c1a990a7d2fb90f070fc93673563c2
> Reviewed-on: https://chromium-review.googlesource.com/c/chromium/src/+/5394656
> Commit-Queue: Michelle Chen <michellegc@google.com>
> Reviewed-by: Darren Shen <shend@chromium.org>
> Reviewed-by: Xiyuan Xia <xiyuan@chromium.org>
> Cr-Commit-Position: refs/heads/main@{#1278661}

Bug: b:327983570, b:331512574
Change-Id: Iad9bc77ae3e993c7840826d228ca7511283737a2
No-Presubmit: true
No-Tree-Checks: true
No-Try: true
Reviewed-on: https://chromium-review.googlesource.com/c/chromium/src/+/5400177
Commit-Queue: Michelle Chen <michellegc@google.com>
Reviewed-by: Darren Shen <shend@chromium.org>
Owners-Override: Tsuyoshi Horo <horo@chromium.org>
Bot-Commit: Rubber Stamper <rubber-stamper@appspot.gserviceaccount.com>
Cr-Commit-Position: refs/heads/main@{#1278756}