Composer install hits errors with scaffold files

After the Drupal 9 upgrade (see video: https://www.youtube.com/watch?v=nLoJ2L0AzDY), I noticed that when running `composer install` I'm getting the following:

```
Class DrupalProject\composer\ScriptHandler is not autoloadable, can not call post-install-cmd script
```

To get the site upgraded in a 2-hour timespan, I kinda cut a few corners tossing out some old `composer.json` cruft, and I seem to have broken some of the stuff that the drupal-project/composer-template had originally injected into the Composer config (like the ScriptHandler... might not even be needed anymore?).

Inspect composer.json and the DrupalProject\composer\ScriptHandler reference first, then compare the current Composer configuration with the scaffold files mentioned in the issue. Confirm which configuration is required after the Drupal 9 upgrade and verify that composer install completes without the autoloadable-class error.

Written by the indexing model from the issue text.