[144] [omnibox][next] Fix classic popup not opening after closing AIM

When closing AIM popup, we share the user's in progress input with the
classic omnibox. Doing so was setting the omnibox state as having input
in progress which prevents zero suggestions.

Classic omnibox opening on click is a side affect of having zero
suggestions to show. In fact, on some sites, where there are no zero
suggestions, classic omnibox doesn't open on click. This used to be
super common, but it's now rare because we have both zero search
suggestions and contextual suggestions.

This CL adjusts the
`OmniboxAimPopupWebUIContent::OnPageClosedWithInput()` code for sharing
user's AIM input with the omnibox to only do so when the AIM input is
not empty. This is consistent with how
`OmniboxEditModel::RestoreState()` does it when switching tabs.

crrev.com/c/7205626 is another example, where `OnPageClosedWithInput()`
needed some logic that was already done in `RestoreState()`. The rest of
the code in `RestoreState()` doesn't seem relevant for
`OnPageClosedWithInput()`, so it doesn't seem like we should leverage
that or a shared helper, but if we continue finding bugs that are
already solved by `RestoreState()`, maybe we should.

(cherry picked from commit 8d206cd3e072c7b8864352fe48012126087115a2)

Bugs: b:463432768, b:463444960, b:462477206, b:462787431, 467113257
Change-Id: I49e52a003fbc101a9f773249244ae26592099aa2
Reviewed-on: https://chromium-review.googlesource.com/c/chromium/src/+/7209782
Reviewed-by: Khalid Peer <khalidpeer@chromium.org>
Commit-Queue: manuk hovanesian <manukh@chromium.org>
Cr-Original-Commit-Position: refs/heads/main@{#1553070}
Reviewed-on: https://chromium-review.googlesource.com/c/chromium/src/+/7240371
Commit-Queue: Justin Donnelly <jdonnelly@chromium.org>
Reviewed-by: manuk hovanesian <manukh@chromium.org>
Cr-Commit-Position: refs/branch-heads/7559@{#607}
Cr-Branched-From: 223dfbac1c7542a06b422390d954afe5b560b607-refs/heads/main@{#1552494}