It’s a great idea. Here’s an add-on that does that.
This tiny change provides a dramatic usability enhancement to the browser. You can just click around on stuff in the sidebar without your delicately crafted search terms vanishing. Huge time saver.
When browser is opened, the sticky search box is in focus, I think it should be the default anki search box in browser.
Also, the sticky search box appears alongside the default one on the same line, it would be better if it appeared on a separate line maybe above it or below it to make it less crowded.