All of the web services listed here have you start by choosing from a selection of templates for your site. The better ones, such as Duda, Gator, Squarespace, and Wix, use templates that automatically reformat your site for viewing on mobile devices. They also offer specifically targeted templates based on your site's purpose, such as for promoting a bakery's sales, getting gigs for a musician, or keeping wedding guests informed.
Gator Website Builder has more than 200 templates that are professionally designed and available for free. And yes, all of them are mobile design compatible. You don't need to do anything to activate the mobile design. Gator does allow you to edit the content in your mobile version without affecting the desktop version. Editing content on the mobile version can be handy if you have special mobile-only sales or if you want to optimize the experience people have on their phone.