To be honest, three nested RDPs sound like a terrible hack. In an ideal world, this would be two port forwardings and one RDP (thinking about ssh, which is still underrepresented in windows world). In an even more ideal world, this would be an IPv6 direct access ;-)
When it’s in a browser you don’t need to install anything on the local machine. I used to use Apache guacamole to access my machine at home from work when I was stuck in a cube all day.
https://guacamole.apache.org/