Automate the declaration of asset dependencies in PHP via bundler

Gutenberg is using a tool to automatically maintain awareness of which asset needs which other assets as dependencies in PHP, using [`DependencyExtractionWebpackPlugin`](https://www.npmjs.com/package/@wordpress/dependency-extraction-webpack-plugin).

In Site Kit we maintain all dependencies manually, which occasionally has led to errors and overall is a very error-prone approach given the amount of JS assets we have. We should consider automating this, similar to how Gutenberg does.

This was originally explored in a draft PR #1521.

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## Acceptance criteria

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## Implementation Brief

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### Test Coverage

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### Visual Regression Changes

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## QA Brief

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## Changelog entry

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Start by reviewing the existing manual asset dependency declarations and the earlier draft PR #1521, then compare the approach with Gutenberg’s DependencyExtractionWebpackPlugin. Done means dependencies are generated automatically for the relevant PHP-registered assets, with the current manual error-prone process replaced or appropriately integrated.

Written by the indexing model from the issue text.