Reland "Add basic logic for positioning the Picker widget near the caret."

This is a reland of commit 1a754aaf4bbf5bc9352e55a32f537cce0e43dcba

The original CL kept a const reference to the screen work area while
computing Picker bounds. This failed sanitizer tests since we don't
know the lifetime of the screen- it could be a temporary fake screen.

To fix this, copy the screen work area instead: see diff between
Patchset 1 and Patchset 3.

(Also make some other changes so that we're not overriding Layout()
to set the Picker widget bounds. See Patchset 4 and onwards.)

Original change's description:
> Add basic logic for positioning the Picker widget near the caret.
>
> - If there is space to the right of the caret, place the Picker widget
>   to the right of the caret and try to vertically align the Picker
>   search field with the caret.
> - Otherwise, place the Picker widget above or below the caret.
>
> Since we want to align the Picker search field with the caret, we only
> position the Picker widget after PickerView layout has happened.
>
> Note that various cases still need to be dealt with (e.g. when there is
> a selection instead of a caret). We'll handle these cases in followups.
>
> Bug: b:320787410, b:322281223
> Change-Id: I7cf427f3c8aa1e229c7a1cdcae1e3939457905aa
> Reviewed-on: https://chromium-review.googlesource.com/c/chromium/src/+/5233749
> Commit-Queue: Michelle Chen <michellegc@google.com>
> Reviewed-by: Darren Shen <shend@chromium.org>
> Cr-Commit-Position: refs/heads/main@{#1251950}

Bug: b:320787410, b:322281223
Change-Id: Ia58e0e58d59eb753532cdb6ad85684cd2f7dcbc4
Reviewed-on: https://chromium-review.googlesource.com/c/chromium/src/+/5247605
Reviewed-by: Darren Shen <shend@chromium.org>
Commit-Queue: Michelle Chen <michellegc@google.com>
Cr-Commit-Position: refs/heads/main@{#1253730}