fix: unintended exception omittion

[shkumagai]

As I wrote in #735, StructuredLogHander removes exception info from the record object directly to omit traceback in a structured log message (it was fixed in #394).
This affects any log handlers that will be called later than StructuredLogHander.

This PR changes implementation to omit traceback in a log message.

• Make sure to open an issue as a bug/issue before writing your code! That way we can discuss the change, evaluate designs, and agree on the general idea
• Ensure the tests and linter pass
• Code coverage does not decrease (if any source code was changed)
• Appropriate docs were updated (if necessary)

Fixes #735 🦕

[daniel-sanche]

thanks for putting this together!