Which GIS (with GUI) produces good-looking maps for a reasonable price?

Most of the GIS packages I have used have excellent mapping tools. I can produce very good maps with QGIS for instance. So that's an option, if you can't run to the cost of ArcGIS. Yes it is a little more fiddly to get the exact result you want but excellent maps are perfectly possible. I am able to produce large maps with QGIS - but you have to sometimes turn off the option to "print as raster". I create them as PDFs and then print the PDF. I don't print direct from QGIS and I never printed directly from ArcGIS for that matter either.

Alternatively you can try MapNik. Its sole purpose is to create maps rather than analysis. Another, more obscure, product that might be worth a look is MapMaker. Another cheap product is Idrisi. These are just a few suggestions.

Given that some of the most beautiful cartography was done by hand using pens and paint, it is worth remembering that it is NOT any particular tool which produces the beautiful map but the cartographer. Map making is as much about art as it is science so the real limitations are artistry of the Cartographer and their imaginative use of whatever tools are available.

I would use QGIS with Illustrator or Inkscape. In QGIS you can convert the data into a form that you can load into Illustrator or Inkscape and there you can manually adjust the image to your preference.

In response to the comment about the data size, you can use MapShaper to reduce how large the raw data source is.