refactor: Enforce iwyu on src/init #36104

This PR continues the ongoing effort to enforce IWYU warnings.

See Developer Notes.

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Code Coverage & Benchmarks

For details see: https://corecheck.dev/bitcoin/bitcoin/pulls/36104.

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🚧 At least one of the CI tasks failed. _{Task i686, no IPC: https://github.com/bitcoin/bitcoin/actions/runs/33095983783/job/98600895025} _{LLM reason (✨ experimental): CI failed due to a C++ build error: bitcoin-wallet.cpp fails to compile because std::unique_ptr is used with incomplete types (interfaces::Node/Chain/...), causing invalid application of sizeof to incomplete type.}

<details><summary>Hints</summary>

Try to run the tests locally, according to the documentation. However, a CI failure may still happen due to a number of reasons, for example:

• Possibly due to a silent merge conflict (the changes in this pull request being incompatible with the current code in the target branch). If so, make sure to rebase on the latest commit of the target branch.

• A sanitizer issue, which can only be found by compiling with the sanitizer and running the affected test.

• An intermittent issue.

Leave a comment here, if you need help tracking down a confusing failure.

</details>

All of those are passing locally for me. Not sure what is different about my setup that this fails CI. Closing for now. Someone else can pick this up.

All of those are passing locally for me. Not sure what is different about my setup that this fails CI. Closing for now. Someone else can pick this up.

Picked in #36107.