At least with Firefox, doing tasks in the background was sometimes quite painful, especially during development. For example, I do some CPU-intensive stuff in the background, close the tab because I noticed a mistake in my code -> background task still runs for several seconds eating CPU and memory until it is killed or finished (I didn't check to be honest) by the browser.

Edit: As with everything related to complex web applications (and I had two or three in my career), it is obvious that the whole environment (client, protocols, languages,...) was never designed for this.