Has anyone heard of a service like Microsoft Mesh except for SSH rather than Remote Desktop?
Normal SSH doesn't work via Web proxy, Mesh does. But I find Remote Desktop too slow and would prefer SSH access to my home network.
PuTTY has a built-in ability to go through a web proxy.
Go to the Connection -> Proxy screen, select HTTP for the Proxy type, and provide the hostname and port. Provide a username and password if required by your proxy.
Some web proxies will refuse to connect you to any port except 443 because they are intended for HTTPS connections. SSH being port 22 normally, this poses a problem. You may consider moving your SSH server to port 443 in this case.