test: fix flaky doctool and test

Doctool tests have been failing a lot in CI on Win2008 R2. It appears
async functions and callback-based functions are being used in
combination such that the callback-based function cannot guarantee that
it will invoke its callback. Convert the callback-based functions to
async functions so we have one paradigm and reliable results.

Backport-PR-URL: https://github.com/nodejs/node/pull/32642
PR-URL: https://github.com/nodejs/node/pull/29979


Reviewed-By: Joyee Cheung <joyeec9h3@gmail.com>
Reviewed-By: Gireesh Punathil <gpunathi@in.ibm.com>
Reviewed-By: Jeremiah Senkpiel <fishrock123@rocketmail.com>