[json-rpc-engine] Fix any types in catch blocks, refactor

[MajorLift]

Explanation

• Fixes all any usage in the JsonRpcEngine class.
• Errors are typed consistently as unknown to reflect actual usage in consumers and tests. - JsonRpcEngineCallbackError is no longer used within the package.

References

• Closes #3720

Changelog

@metamask/json-rpc-engine

• BREAKING: The error types used in JsonRpcEngineReturnHandler, JsonRpcEngineNextCallback, and JsonRpcEngineEndCallback are widened from JsonRpcEngineCallbackError to unknown.

Checklist

• [x] I've updated the test suite for new or updated code as appropriate
• [x] I've updated documentation (JSDoc, Markdown, etc.) for new or updated code as appropriate
• [x] I've highlighted breaking changes using the "BREAKING" category above as appropriate