I think you are asking whether what you call the pop approach to improvising—finding a common scale/key that works over many chords in sequence—is valid in jazz. The simple answer is yes. Indeed, learning to spot common scales and tones is an important hurdle that many musicians trained to think chord by chord learn to overcome in college.
In the example you've given, a ii-V7-I in C major, note that the scales you derive when thinking about the changes individually actually consist of the same collections of notes:
- Dm will be a D dorian, which contains the same notes as C major
- G7 will be G mixolydian, also equivalent to C ionian.
- Over the C, you're actually likely to use the ionian mode in this context. (The
lydian may be used if you have an extended vamp on this passage or
for a more "modern" sound, but it's perfectly valid to use ionian as
long as you don't attempt to resolve on the F.)
Thus, if you are simply looking for a collection of notes that works well over all these chords, then C ionian (C major) will certainly do the trick. Of course, jazz songs are more likely than pop songs to switch key centers quickly, but you can still use the same approach. If you have a chord progression like
Dm G7 C %
Fm Bb7 Eb %
you can play C major over the first four bars and, by similar logic, Eb major over the next four bars.
This is called the key-centers approach.
If the key-centers approach is valid, why do books on jazz improvisation often ask you to think about the chords individually (which is called the modal approach)? Two reasons, both (in my opinion) pedagogical.
First, beginner musicians tend to be biased towards the first few notes in a scale. If you tell a student that the C major scale is valid over the progression Dm G7 C, they will often just repeat patterns involving the notes C, D, and E. Asking the student to think about these as three different chords guides them to other parts of the scale. The same student, thinking about G mixolydian over the G7 chord, now will center their playing around G, A, and B instead. This is still less than perfect, of course, but it's a teaching strategy designed to encourage students to explore the full range of the scale.
Second, and related, is that while all the notes in the C major scale are broadly compatible with these chords, the color and stability of the individual notes depends on what chord you're on. The note F sounds great over a Dm chord; not so much over a C major (although if you move quickly down to E it's fine). When you think about the modes separately, you may be able to spot the stable notes—like 1, 3, and 5—more quickly.