On Big Sur 11.7.3 two updates are available: macOS 11.7.4 and Safari 16.3.1. Because 11.7.3 update had caused me a bunch of problems, I decided to install only the Safari update. Curiously the installer didn’t ask to quit Safari as normally expected. In fact after the installation nothing changed in Safari’s version number: Version 16.3 (16614.4.6.11.6, 16614) even after relaunching Safari.
Everything you said happened to me, but with more. The 11.7.4 update pane also offered Safari 16.3, which, to that level of detail, is what I had. Something around the declared size of the Safari update was downloaded, and the update proceeded normally (which means it took about 50 minutes).
Immediately after the installation, System Preferences sported a badge with a 1 in it. Minutes of “Checking for updates…” had no effect; it was not until after I restarted that Software Update was able (in short order) to offer Safari 16.3.1.
It’s like keeping system software up to date is a game of hide-and-seek or a treasure hunt. And then the prize is copyright 2022.