[mcp] Fix package.json (#33101)

Since we use esbuild we need to correctly move dependencies that are
required at runtime into `dependencies` and other packages that are only
used in development in to `devDependencies`. This ensures the correct
packages are included in the build.
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[//]: # (BEGIN SAPLING FOOTER)
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* #33085
* #33084
* #33083
* #33082
* __->__ #33101