Clean up the Search block's controls

[octaedro]

What problem does this address?

WooCommerce issue #47890 requested some changes, and as a follow-up of PR #51247 it would be good to apply some changes in core components.

Most of the Search block's layout-related settings live in the toolbar, while there's nothing in the Inspector. The toolbar is supposed to house only the most commonly used tools and it's unexpected to see such options there.

What is your proposed solution?

We can move the commonly used tools to the Inspector and clean up the UI to make designing the perfect product search field easier.

// CC @jarekmorawski @j111q

[Marc-pi]

Related to #64988 also (see @jasmussen's comments over there)