You don't need to specify a given port to run software on a different computer with SSH, if that's what you're saying. You could run Firefox on a remote system by simply running a command like this:

ssh -X hostname firefox

You don't even need the -X if you have SSH configured to automatically forward the X Window System protocol -- though that's usually a bad idea for security reasons (needlessly forwarding protocols can serve as a security exposure -- including using something like Remote Desktop or VNC when all you actually need to do is run shell commands).

One of the reasons you might want to run a local instance of Firefox that uses an encrypted tunnel to a remote proxy is to reduce the bandwidth demands of the remote connection -- and, thus, reduce problems like latency and bandwidth bottlenecks. Another is to make it easier to interact with the local system, such as when downloading a file.

Remember that your HTTP stream has to travel across the Internet which, even with a broadband connection, usually means a much more restrictive bandwidth connection than on a local network. On the other hand, adding the massive bandwidth overhead of forwarding an entire application interface across the Internet to run it on a computer at home from some coffee shop is just begging for tears of frustration.

"Installing the client is simple and firewall porting is not required."

If, by "firewall porting", you mean port forwarding on the local system to connect to the secure proxy, you're obviously talking about not using a secure proxy setup on the local machine. Instead, you would have to spend time setting up a proxy server on the remote system (which is a much more complex process than running that simple SSH command in the article) or, more likely based on what I think you're suggesting for LogMeIn use, running a remote instance of a browser (in which case my above explanation of why a proxy is better comes into play).