I’ve been looking for good solutions to this same problem myself.
The 404 error you receive when navigating to /products/ is actually normal. You can see another example of this error by trying to navigate to /category/, which is a built in taxonomy in WordPress.
So far the best solution I have come across is to create a new page in WordPress with the same name as the slug of your Taxonomy (in your case, ‘products’). You can then assign that page a new page template. In the template you can then build your own query to output index content. That seemed to do the trick for me.
Seems counterintuitive considering that with Custom Post Types you can pass the has_archive parameter, but this is not that case with taxonomies.
