You can accomplish by checking the "Automatically download updates like this in the future" box in Sparkle. The problem, though, is that you have to trust the software to handle these updates perfectly if you want them to happen in the background for critical software. Chrome can do that, because Google has some of the best developers around working on the project, and because they have absurdly good development practices. (Not perfect, but far better than any independent developer or small shop working on the majority of apps that we're talking about).
Automatic updating is hard. About a year ago, there was an update to iTerm2 that broke the software completely. The fix was to temporarily downgrade while waiting for the next point release - but for people who allowed automatic updates, they simply discovered one morning that they couldn't open their terminal[0].
Slack had an issue as well about a year ago as well, though that issue was with their auto-updater itself being broken, and preventing them from upgrading. In that case, they had to request that affected users uninstall and reinstall the application, just to get the new version of the automatic updater installed[1].
[0] I think this was the one: https://gitlab.com/gnachman/iterm2/issues/4699
[1] It's hard to push an update to your software automatically if your automatic updater is what's broken.
The only one I can think of that makes a copy is the one from Omni Group, which seems to leave an old version in the Trash. Plus, it renames the application so it has an exact “version” for clarity.
I've never used the mac version, but Squirrel for windows works great and is unobtrusive. The Mac version promises the same: "Squirrel is an OS X framework focused on making application updates as safe and transparent as updates to a website.".
Its a total momentum killer. Like, imagine if randomly (every month or so) instead of loading immediately hackernews took 1 minute to load instead, and it required that you babysit it the whole time while you hit "next".
Personally I wish all apps used Chrome's update mechanism. Its unobtrusive, the patches are tiny compared to the size of Chrome and it works great. Most users don't even know the updates are happening.
Thats how I want all apps (and, frankly, my OS) to update. Just, silently and reliably. No annoying popups. No decisions to make when I'm trying to get work done. No waiting for the update to happen. Update yourself quickly in the background, and do it without bothering me.
Given how long chrome's updater has been out I'm surprised there's no embeddable Sparkle-like version for 3rd party apps to use.