The problem with these online site builders is that they are much like Wordpress in that they generate huge amounts of code for even a simple page.
Smart phone users are often depending on shaky cell phone connections and a lot of them are up against data limits.
Creating a mobile version of a site is fairly straightforward using
jQuery Mobile. The help file is comprehensive. You can convert it for use on desktops by centering the layout and setting a max-width.
Customizing themes is done using the online theme roller.
If you want to create a "one size fits all" website,
Initializr is a good starting point for getting into responsive design. I use an adaption of this for sidebar and column layouts using blocks to get a better layout on tablets.