The map shows nothing on the page
Make sure the map is published (not a draft), and that you selected it in the block or used the correct id in the shortcode. In the block editor, a draft map previews for you but stays hidden from visitors until you publish it. A published map always renders, falling back to a default US states map if it has no saved configuration yet.
My colours do not change when I edit the site-wide settings
Site-wide colours only apply to areas that have not been individually overridden. If you set a colour on a specific area in the map editor, that area keeps its own colour and ignores the global palette. Clear the area's colour fields to make it inherit again. Maps imported from the legacy plugin keep the colours your old map had set.
I changed something but the front end looks the same
Your browser or a caching plugin may be serving an old copy. Hard-refresh the page (Ctrl+Shift+R), and purge your page cache and any CDN if you use one.
The legacy import did not run
Auto-import runs once on activation, only when the old US Map or US Regional Map option data is present on the site. If you added that data later, open any map and use the Tools panel to import on demand. Deleting an imported map releases its source so it can be imported again.
Is the front end really dependency-free?
Yes. There are no external requests at all: no tiles, no map API, no fonts, no trackers. You can confirm in your browser's network tab that the only request is the plugin's own small CSS and JS, and only on pages that display a map.