The venerable Netscape plug-in API is about as browser-agnostic as one can get; I don't think there's a modern GUI browser that doesn't support it, except perhaps Amaya or something.
At least, this is my impression. The line gets fuzzier all the time, so feel free to disagree.
Opera has it built in. IE has http://www.gm4ie.com/ . WebKit has http://8-p.info/greasekit/ . Chrome http://greasemetal.31tools.com/ and will likely have it built-in. And, of course, FireFox has GreaseMonkey itself.
But the author wants some kind of proxy that works for all browsers, even if they don't have GM support or even anything like plugin support.
I'll add an update to the blog post to clarify.