Revert "[OmniboxEverywhere] Extract public target for controller headers"

This reverts commit bb5958d38c35cd0a10877dbea91cf16206201cad.

Reason for revert:
LUCI Bisection has identified this change as the culprit of a build failure. See the analysis: https://ci.chromium.org/ui/p/chromium/bisection/compile-analysis/b/8674294551664958577

Sample failed build: https://ci.chromium.org/b/8674294551664958577

If this is a false positive, please report it at http://b.corp.google.com/createIssue?component=1199205&description=Analysis%3A+https%3A%2F%2Fchromium-review.googlesource.com%2Fc%2Fchromium%2Fsrc%2F%2B%2F8191319&format=PLAIN&priority=P3&title=Wrongly+blamed+https%3A%2F%2Fci.chromium.org%2Fui%2Fp%2Fchromium%2Fbisection%2Fcompile-analysis%2Fb%2F8674294551664958577&type=BUG

Original change's description:
> [OmniboxEverywhere] Extract public target for controller headers
>
> Bug: None
> Change-Id: Ib60aead49ac10d5367751da799134aca4b6a8ff0
> Reviewed-on: https://chromium-review.googlesource.com/c/chromium/src/+/8191319
> Reviewed-by: Juliet Lévesque Knighton <julietlevesque@google.com>
> Commit-Queue: Atharv Maan <atharvmaan@chromium.org>
> Cr-Commit-Position: refs/heads/main@{#1674197}
>

Bug: None
No-Presubmit: true
No-Tree-Checks: true
No-Try: true
Change-Id: I2b4f54c623d9a21faadd555609f660008d7233eb
Reviewed-on: https://chromium-review.googlesource.com/c/chromium/src/+/8203245
Owners-Override: luci-bisection@appspot.gserviceaccount.com <luci-bisection@appspot.gserviceaccount.com>
Commit-Queue: luci-bisection@appspot.gserviceaccount.com <luci-bisection@appspot.gserviceaccount.com>
Bot-Commit: luci-bisection@appspot.gserviceaccount.com <luci-bisection@appspot.gserviceaccount.com>
Cr-Commit-Position: refs/heads/main@{#1674200}