There are several ways and services that already exist. One more would certainly be welcomed, can't guarantee it will catch on.
http://people.apache.org/~pmuellr/weinre-docs/latest/Home.ht...
http://jsconsole.com/remote-debugging.html
The examples might show debugging on a local network, but it is certainly possible (and fairly easy) to do it across the internet.
I would advise against running it on a site with even a moderate amount of traffic.
This might also have some features that you are looking for: http://newrelic.com/real-user-monitoring They're not free though.
The steps to use these tools are generally: insert a single javascript file (with a unique identifier) into the pages you want to debug, the view their custom console for errors. It's very simple and very useful.