Shutting down the software was to prevent the BDB environment from writing unencrypted private keys to disk in the database log files, as was noted here. This PR replaces the shutdown behavior with a CDBEnv flush, close, and reopen which achieves the same effect: everything is cleanly flushed and closed, the log files are removed, and then the environment reopened to continue normal operation. To ensure that no unencrypted private keys are in the log files after encrypting the wallet, I wrote this script to pull private keys from the original wallet file and searches for these keys in the log files (note that you will have to change your file paths to make it work on your own machine). As for concerns about private keys being written to slack space or being kept in memory, these behaviors no longer exist after the original wallet encryption PR and the shutting down solution from 2011.
Backport of core PR12493: