To use WhatsApp Web, your actual mobile device still needs to be connected. If I completely disconnect my phone, e.g. put it in airplane mode, I get this on WhatsApp web: Phone not connected. Make sure your phone has an active Internet connection.
The link follows to https://faq.whatsapp.com/faq/web/28080002
So I admit, I'm also struggling to see what the benefit of using WhatsApp through Browserling is.