[148] [lens] Delay searchbox initialization to fix flaky thumbnail

Original change's description:
> [lens] Delay searchbox initialization to fix flaky thumbnail
> 
> Move the OnPageBound() invocation from the LensSearchboxHandler
> constructor to the LensSearchboxController registration methods.
> 
> Previously, OnPageBound() fired during the handler object's
> construction. At this stage, the controller's pointer to the handler was
> still null, causing initial state flushes, such as cached thumbnails and
> pending text queries, to be skipped.
> 
> Delaying this notification until the handler is fully stored in the
> controller ensures that the initial state push always succeeds. This
> resolves the flaky viewport thumbnail issue by securing the existing
> state propagation order without introducing side effects.
> 
> Demo: http://shortn/_DGK0HD8ARP
> Bug: b:510419641
> Change-Id: If0abf11b601f78f5555969295e224cec30a82162
> Reviewed-on: https://chromium-review.googlesource.com/c/chromium/src/+/7830897
> Auto-Submit: Juan Mojica <juanmojica@google.com>
> Commit-Queue: Juan Mojica <juanmojica@google.com>
> Reviewed-by: Riley Tatum <rtatum@google.com>
> Commit-Queue: Riley Tatum <rtatum@google.com>
> Reviewed-by: Duncan Mercer <mercerd@google.com>
> Cr-Commit-Position: refs/heads/main@{#1627226}

Bug: 512080682
Change-Id: If0abf11b601f78f5555969295e224cec30a82162
Reviewed-on: https://chromium-review.googlesource.com/c/chromium/src/+/7836110
Bot-Commit: rubber-stamper@appspot.gserviceaccount.com <rubber-stamper@appspot.gserviceaccount.com>
Commit-Queue: Justin Donnelly <jdonnelly@chromium.org>
Cr-Commit-Position: refs/branch-heads/7778@{#2783}
Cr-Branched-From: 77f495ee216d4c3cc784d33658bad4778c0680ee-refs/heads/main@{#1610480}