Gaia sign-in screen: Remove usage of Polymer's $$

Remove the usage of Polymer's |this.$$()| in favor of using standard
DOM methods.

The Polymer's |$$| method is deprecated and is removed in Polymer 2.0:

https://polymer-library.polymer-project.org/2.0/docs/upgrade#dollardollar

Bug: 964069
Change-Id: Id0725dbd427cf58a6ec9dba3207bf012864206ce
Reviewed-on: https://chromium-review.googlesource.com/c/chromium/src/+/1627376
Commit-Queue: Maksim Ivanov <[email protected]>
Reviewed-by: Alexander Alekseev <[email protected]>
Reviewed-by: Roman Sorokin [CET] <[email protected]>
Reviewed-by: Denis Kuznetsov <[email protected]>
Cr-Commit-Position: refs/heads/master@{#664276}
1 file changed