refactor: migrate deprecated $app/stores to modern $app/state

[insign]

Migrates deprecated $app/stores import to $app/state in the main page component.

The previous implementation relied on $app/stores which is deprecated in Svelte 5 and SvelteKit 2.12+. This change updates the page to use import { page } from '$app/state' and removes the $ prefix from all page references, seamlessly integrating with Svelte 5's new reactivity model (untrack, etc.) and resolving potential linter warnings for reactive tracking outside boundaries. No functional changes have been introduced; this is purely a modernization to support long-term SvelteKit maintainability.

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PR created automatically by Jules for task 4386353796072929764 started by @insign

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