Emoji picker: Make handler variables private

Other classes probably shouldn't be touching these variables, so make them private

Change-Id: I6690325a3b7d4764c79267dd738047d9f089dfbc
Reviewed-on: https://chromium-review.googlesource.com/c/chromium/src/+/5465412
Auto-Submit: John Palmer <jopalmer@google.com>
Reviewed-by: Zac Partridge <zacpartridge@google.com>
Commit-Queue: Zac Partridge <zacpartridge@google.com>
Cr-Commit-Position: refs/heads/main@{#1290520}