For future travelers. You can add entries via Wikimedia:Sidebar, but the current code has an apparent bug, maybe, introduced around 1.35 or with changes for sidebar cache.
$sidebar['TOOLBOX'] = $this->makeToolbox(
$this->buildNavUrls(),
$this->buildFeedUrls()
);
$sidebar['LANGUAGES'] = $this->getLanguages();
Should likely be:
$sidebar['TOOLBOX'] = array_merge($sidebar['TOOLBOX'], $this->makeToolbox(
$this->buildNavUrls(),
$this->buildFeedUrls()
));
$sidebar['LANGUAGES'] = array_merge($sidebar['LANGUAGES'], $this->getLanguages());
See: mediawiki-1.36.0-wmf.31/includes/skins/Skin.php
This would unfortunately mean editing core mediawiki.
Just above in the cache callback stub, you can see it call addToSidebar(), which reads the Wikimedia:Sidebar. then in the code i posed above it gets discarded. It is actually reading the TOOLBOX section and doing the work, just throwing it away i presume by accident.
$bar = [];
$this->addToSidebar( $bar, 'sidebar' );
$this->getHookRunner()->onSkinBuildSidebar( $this, $bar );