If Software Update is not offering you a newer version of Safari, it is because it doesn't think there is a newer version that will run on 10.14.6. Have you tried opening the Apple App Store to search for any version there?
Have you tried using any other Browsers? If you want to avoid Chrome/Mozilla/etc., you might try DuckDuckGo or several other free browsers that may run on your OS.
BTW, don't believe everything Apple tells you, you do not have to install "OS15" to run Safari 15.6.1. I'm running it just fine with
Big Sur macOS
11. In fact, my wife's MBP uses that version of Safari in
Catalina macOS 10.
15! All Apple is trying to do is to sell you a new machine or two.
Not sure why you didn't get Safari 14 with Mojave!
Safari 14 for MacOS ... Mojave Released (OSXDaily, Sep 16, 2020).