Genuinely curious: when you want a "cross-browser extension", why wouldn't a web-app do? Isn't an extension specifically something that requires non-standard browser APIs and hence can't be made part of a website?
while the actual work of the extension may be done in a web app, extensions get more access to the interface of a browser than a normal web app
example: the Buffer extension could probably just be a button that simply loads a JS file from their servers that does all the real work. WIthout being an extension, you wouldn't have the convenience of having the button on every page you view.