The only really major difference between Giant Steps and any other tune that came before it is that the keys change by major thirds and it's really fast. It doesn't really demand anything different of a player except for a lot of practice to get used to the fast and comparably awkward key changes.
IMO passing tones aren't just key in Giant Steps, they're key in every single tune you could play. Every good piano, guitar, sax player, etc, uses them as the basis for ornamenting the chord changes. I'm a big proponent of practicing arpeggios with a half step above, below, above then below, and below then above each chord tone, and then do it with all the tensions.
Don't get me wrong, it's important to know and practice your modes. And modes actually really are very useful in soloing when it comes to altered dominant chords. But I think your example of "G mixolydian over G7" is lacking because you really do have to avoid the 4/11. And I listen to students practicing, and they're stepping all over that 4. That's a case in point why I think making modal playing the focus of improvisation is seriously flawed in functional harmony. When it comes down to it, I think just thinking about the key center and where keys change is the best way to go about it (**in harmonically simple tunes**), because you can ornament and flip the major scale to be anything you want, and you're not burdened by thinking about altering each dorian, phrygian, lydian, etc. By using the key as the focus, you can become much more fluent in your playing. Plus, just about any arpeggio within the major scale is going to fit with any chord in the key.
YMMV.