New mobile menu implementation

[Blargian]

Summary

As delicious as double burger menu sounds, it should not be in a navigation menu...
new implementation of the mobile menu on docs.

• Single burger menu
• Search is moved outside of the menu
• We show both the main menu and the side bar in the same menu
• Various tweaks and tuning to css to make the site more responsive

Checklist

• Delete items not relevant to your PR
• URL changes should add a redirect to the old URL via https://github.com/ClickHouse/clickhouse-docs/blob/main/docusaurus.config.js
• If adding a new integration page, also add an entry to the integrations list here: https://github.com/ClickHouse/clickhouse-docs/blob/main/docs/integrations/index.mdx

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Name	Status	Preview	Comments	Updated (UTC)
clickhouse-docs	✅ Ready (Inspect)	Visit Preview		Jun 9, 2025 0:13am

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clickhouse-docs-ru	⬜️ Ignored (Inspect)	Visit Preview		Jun 9, 2025 0:13am
clickhouse-docs-zh	⬜️ Ignored (Inspect)	Visit Preview		Jun 9, 2025 0:13am

[DamianMaslanka5]

FYI, kubernetes.io also has two menus, IMO it looks good