On Windows, I could use Tab to shift focus from button to button, so I could "click" on the desired button without having to use a mouse.

It appears that I can't do the same thing on a Mac, or is there a way to do it that I didn't realize?

There is. From System Preferences, under Keyboard ยป Keyboard Shortcuts, you can enable Full Keyboard navigation. You can also change this by pressing Control+F7