Put the extra tools in the "footer" of the navigation sidebar

This is a great point. I agree, the tools would see to fit better with the items in the footer than the rest of the navigation bar.

I think it would be nice if it was something the site builder can decide on, so adding a configuration to allow this would be useful.

I'm going to focus on getting a 1.0.0 stable release, then I will look at this.

As this is a new feature, I'm flagging it for consideration for version 1.1.

Potentially related core issue: ✨ Allow modules to hook into top of content/footer sections of new core navigation Active

I discussed on the #admin-ui channel on Drupal Slack in this thread.

At present the Navigation do not want links added to the footer menu, and consider it a "constrained space", and I don't want to disrespect that.

I'd appreciate if anyone with interest could read the thread above and add any thoughts to this issue.

Just pitching in here, we now have navigation_extra_tools support in our navigation_extra module. It allows you to place the links under the "drupal" logo as it was with the toolbar version. We can add the option so you can put them under the "user" menu too.
Our navigation_extra module is currently still in dev only, but we are working on a stable release.