It's definitely a lot more work to set up, but it's definitely possible. Writing raw AppleScript scripts and using some other method to trigger those when you need to is going to be as factory-Apple as you can.

You'd then be able to click on it and see each script as those they were individual buttons/sub-options within the TextBar menu app icon. You'd just have to write an AppleScript script to trigger each unique screenshot action and then pipe those into TextBar. If you absolutely 100%, absolutely had to trigger the bundled Screenshot.app, then your best bet is either a tiny menu bar app like TextBar that allows you to write your own fancy scripts and popouts to monitor and trigger anything you want. Now you get a better screenshotting experience without complicated how you take each type of screenshot. So all you would need to do is unassign the default actions within System Preferences and reassign them within your new app. If you're just worried about the specific hotkey combos as you don't want to learn new ones, that's perfectly understandable, but nearly every external app allows you to set whatever trigger you want. But there are quality free options like Shottr. If it's just because you don't need all the fancy capture and annotation options, then sure, you don't need something like CleanShotX as it's probably too feature-rich and too expensive for your needs. I'm trying to understand your exact aversion to using a third-party screenshot app, especially if you're willing to still install some sort of non-Apple app to control the functionality.

Apple mac screenshot shortcut