There are a few requirements that should be clarified. Will the clients build websites in their own domain (name) or will the sites be subsites of your domain? Are you a programmer or are willing to pay a bit for a programmer if necessary? It is likely you can find solutions that do not require programming but the option to do programming can increase the possibilities.
In my opinion CMSs are more like website builders than anything else. In The Best Free and Open Source Website Builder Software the What are the essential Elements of the Website Builder? lists CMS plus everything else listed there I think are features of CMS systems such as Wordpress. And Wordpress can be rebranded.
There are likely too many options. People can make recommendations based on their requirements that would not be good for you. I think we should get clarification of requirements before attempting to speculate on what would help.
Probably an important reason to not use that is that it is proprietary and difficult for customers to convert to another service if they need to or just want to. And the client might need to pay for service and if so then (some) of the revenue goes to Weebly. Are you sure that the website builder can be rebranded?