Docs landing page revamp

[Blargian]

The docs landing page needs some love. Incorporate the feedback from design:

• The search dialog is showing a dark blue background — it should use colors to mimic our standard dark theme colors. ?
• The icons on Get Started, Learn, and Reference don’t work well as illustrations (they’re also pixelated). Removing them might make the layout cleaner. -> https://github.com/ClickHouse/design/issues/1084
• There’s low contrast between the Integrations square and the Migrate background color.
• The Feature badges should use capital case instead of ALL CAPS.
• The body text on the Feature cards should use a muted white/gray color for better readability.
• For cards in general, we could add a light gray stroke to match the styling of the menu items.
• The page width is currently ~980px. We might want to consider increasing it slightly.
• Search and Ask AI look connected, almost as if the user should type and then click Ask AI to “search” (not intended). Consider adding an “or” between search and Ask AI.
• Spacing should be the same for the list items in "reference" as it is for "getting started" and "learn".
• Use 14px or 16px and font Inter. Consistency of heading sizes between cards.
• Possibly add both Cloud quick start + OSS quick start links to "Getting started" list. It wasn't clear why Cloud getting started is linked in addition to ClickHouse Cloud (get started section) as well.
• Example datasets -> “Sample datasets” (to imply they can be downloaded).
• Styling of cards, buttons, links etc consistent with the website.
• Consider making the migrations section span 3 columns rather than 2. The changelogs could maybe just be text with a small icon, not needed to display them so prominently.