Can a launcher permit to skip Funamo?
A launcher may affect application control but it should not affect web filtering. Are you using Funamo browser? It is not clear from your question when you say "the browser of the phone". I am assuming that it is the built-in browser app (for Samsung, it is called Internet). Web filtering would not work unless you did manual proxy setup with Advanced mode. Please make sure that you use Funamo browser and enable Block other browsers in Account settings so other browsers cannot be used.
More details can be found at instruction page: https://www.funamo.com/android-parental-control-instruction#easysetup