Well you could support both or make a browser extension that simply redirects there on click.
Not sure what permissions are needed to get the current url from the plugin though.
But if it's really more about a good way to browse the way back machine I think a web-service/site would make more sense. Installing a browser extension is quite a hurdle for something you don't need that often.
Too bad bookmarklets are not really a thing anymore, that would work well.
