Instant Search: address browser privacy settings stripping the $query= value. (#45533)

* Attempt to address privacy settings breaking search

Instead of reading from the input event, lets use the inputRef as it should already be available.

* changelog

* Also make change to DomEventHandler

We need to make the same privacy change to DomEventHandler.

Committed via a GitHub action: https://github.com/Automattic/jetpack/actions/runs/18684216649

Upstream-Ref: Automattic/jetpack@6067d66

Author: tbradsha